zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

cos^3(x)= 1/(4(3cos(x)+cos(3x)))

解答

cos^{3} (x) = \frac{1}{4 ( 3 cos ( x ) + cos ( 3 x ) )}

解答

x = 0.88929 \dots + 2 πn, x = 2 π - 0.88929 \dots + 2 πn, x = 2.25229 \dots + 2 πn, x = - 2.25229 \dots + 2 πn

+1

度数

x = 50.95278 \dots^{\circ} + 36 0^{\circ} n, x = 309.04721 \dots^{\circ} + 36 0^{\circ} n, x = 129.04721 \dots^{\circ} + 36 0^{\circ} n, x = - 129.04721 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos^{3} (x) = \frac{1}{4 ( 3 cos ( x ) + cos ( 3 x ) )}

两边减去

\frac{1}{4 ( 3 cos ( x ) + cos ( 3 x ) )}

cos^{3} (x) - \frac{1}{4 ( 3 cos ( x ) + cos ( 3 x ) )} = 0

化简

cos^{3} (x) - \frac{1}{4 ( 3 cos ( x ) + cos ( 3 x ) )} : \frac{4 cos ^{3} ( x ) ( 3 cos ( x ) + cos ( 3 x ) ) - 1}{4 ( 3 cos ( x ) + cos ( 3 x ) )}

cos^{3} (x) - \frac{1}{4 ( 3 cos ( x ) + cos ( 3 x ) )}

将项转换为分式:

cos^{3} (x) = \frac{cos ^{3} ( x ) 4 ( 3 cos ( x ) + cos ( 3 x ) )}{4 ( 3 cos ( x ) + cos ( 3 x ) )}

= \frac{cos ^{3} ( x ) \cdot 4 ( 3 cos ( x ) + cos ( 3 x ) )}{4 ( 3 cos ( x ) + cos ( 3 x ) )} - \frac{1}{4 ( 3 cos ( x ) + cos ( 3 x ) )}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{cos ^{3} ( x ) \cdot 4 ( 3 cos ( x ) + cos ( 3 x ) ) - 1}{4 ( 3 cos ( x ) + cos ( 3 x ) )}

\frac{4 cos ^{3} ( x ) ( 3 cos ( x ) + cos ( 3 x ) ) - 1}{4 ( 3 cos ( x ) + cos ( 3 x ) )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

4 cos^{3} (x) (3 cos (x) + cos (3 x)) - 1 = 0

使用三角恒等式改写

- 1 + (cos (3 x) + 3 cos (x)) \cdot 4 cos^{3} (x)

cos (3 x) = 4 cos^{3} (x) - 3 cos (x)

cos (3 x)

使用三角恒等式改写

cos (3 x)

改写为

= cos (2 x + x)

使用角和恒等式:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2 x) cos (x) - sin (2 x) sin (x)

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

= cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

化简

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x) : cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

2 sin (x) cos (x) sin (x) = 2 sin^{2} (x) cos (x)

2 sin (x) cos (x) sin (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= 2 cos (x) sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= 2 cos (x) sin^{2} (x)

= cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

= cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

= cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

使用倍角公式:

cos (2 x) = 2 cos^{2} (x) - 1

= (2 cos^{2} (x) - 1) cos (x) - 2 sin^{2} (x) cos (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= (2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x)

乘开

(2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x) : 4 cos^{3} (x) - 3 cos (x)

(2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x)

= cos (x) (2 cos^{2} (x) - 1) - 2 cos (x) (1 - cos^{2} (x))

乘开

cos (x) (2 cos^{2} (x) - 1) : 2 cos^{3} (x) - cos (x)

cos (x) (2 cos^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = cos (x), b = 2 cos^{2} (x), c = 1

= cos (x) 2 cos^{2} (x) - cos (x) 1

= 2 cos^{2} (x) cos (x) - 1 cos (x)

化简

2 cos^{2} (x) cos (x) - 1 \cdot cos (x) : 2 cos^{3} (x) - cos (x)

2 cos^{2} (x) cos (x) - 1 cos (x)

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

2 cos^{2} (x) cos (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (x) cos (x) = cos^{2 + 1} (x)

= 2 cos^{2 + 1} (x)

数字相加:

2 + 1 = 3

= 2 cos^{3} (x)

1 \cdot cos (x) = cos (x)

1 cos (x)

乘以:

1 \cdot cos (x) = cos (x)

= cos (x)

= 2 cos^{3} (x) - cos (x)

= 2 cos^{3} (x) - cos (x)

= 2 cos^{3} (x) - cos (x) - 2 (1 - cos^{2} (x)) cos (x)

乘开

- 2 cos (x) (1 - cos^{2} (x)) : - 2 cos (x) + 2 cos^{3} (x)

- 2 cos (x) (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - 2 cos (x), b = 1, c = cos^{2} (x)

= - 2 cos (x) 1 - (- 2 cos (x)) cos^{2} (x)

使用加减运算法则

- (- a) = a

= - 2 \cdot 1 cos (x) + 2 cos^{2} (x) cos (x)

化简

- 2 \cdot 1 \cdot cos (x) + 2 cos^{2} (x) cos (x) : - 2 cos (x) + 2 cos^{3} (x)

- 2 \cdot 1 cos (x) + 2 cos^{2} (x) cos (x)

2 \cdot 1 \cdot cos (x) = 2 cos (x)

2 \cdot 1 cos (x)

数字相乘:

2 \cdot 1 = 2

= 2 cos (x)

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

2 cos^{2} (x) cos (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos^{2} (x) cos (x) = cos^{2 + 1} (x)

= 2 cos^{2 + 1} (x)

数字相加:

2 + 1 = 3

= 2 cos^{3} (x)

= - 2 cos (x) + 2 cos^{3} (x)

= - 2 cos (x) + 2 cos^{3} (x)

= 2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x)

化简

2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x) : 4 cos^{3} (x) - 3 cos (x)

2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x)

对同类项分组

= 2 cos^{3} (x) + 2 cos^{3} (x) - cos (x) - 2 cos (x)

同类项相加:

2 cos^{3} (x) + 2 cos^{3} (x) = 4 cos^{3} (x)

= 4 cos^{3} (x) - cos (x) - 2 cos (x)

同类项相加:

- cos (x) - 2 cos (x) = - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= - 1 + 4 cos^{3} (x) (4 cos^{3} (x) - 3 cos (x) + 3 cos (x))

4 cos^{3} (x) (4 cos^{3} (x) - 3 cos (x) + 3 cos (x)) = 16 cos^{6} (x)

4 cos^{3} (x) (4 cos^{3} (x) - 3 cos (x) + 3 cos (x))

同类项相加:

- 3 cos (x) + 3 cos (x) = 0

= 4 \cdot 4 cos^{3} (x) cos^{3} (x)

数字相乘:

4 \cdot 4 = 16

= 16 cos^{3} (x) cos^{3} (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos^{3} (x) cos^{3} (x) = cos^{3 + 3} (x)

= 16 cos^{3 + 3} (x)

数字相加:

3 + 3 = 6

= 16 cos^{6} (x)

= - 1 + 16 cos^{6} (x)

- 1 + 16 cos^{6} (x) = 0

用替代法求解

- 1 + 16 cos^{6} (x) = 0

令

: cos (x) = w

- 1 + 16 w^{6} = 0

- 1 + 16 w^{6} = 0 : w = \frac{2 ^{\frac{2}{3}}}{2}, w = - \frac{2 ^{\frac{2}{3}}}{2}, w = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, w = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i, w = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, w = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

- 1 + 16 w^{6} = 0

将

1

到右边

- 1 + 16 w^{6} = 0

两边加上

1

- 1 + 16 w^{6} + 1 = 0 + 1

化简

16 w^{6} = 1

16 w^{6} = 1

两边除以

16

16 w^{6} = 1

两边除以

16

\frac{16 w ^{6}}{16} = \frac{1}{16}

化简

w^{6} = \frac{1}{16}

w^{6} = \frac{1}{16}

用

u = w^{2}

和

u^{3} = w^{6}

改写方程式

u^{3} = \frac{1}{16}

解

u^{3} = \frac{1}{16} : u = 3 \frac{1}{16}, u = 3 \frac{1}{16} \frac{- 1 + 3 i}{2}, u = 3 \frac{1}{16} \frac{- 1 - 3 i}{2}

u^{3} = \frac{1}{16}

对于

g^{3} (x) = f (a)

解为

g (x) = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

u = 3 \frac{1}{16}, u = 3 \frac{1}{16} \frac{- 1 + 3 i}{2}, u = 3 \frac{1}{16} \frac{- 1 - 3 i}{2}

u = 3 \frac{1}{16}, u = 3 \frac{1}{16} \frac{- 1 + 3 i}{2}, u = 3 \frac{1}{16} \frac{- 1 - 3 i}{2}

代回

u = w^{2}

，求解

w

解

w^{2} = 3 \frac{1}{16} : w = \frac{2 ^{\frac{2}{3}}}{2}, w = - \frac{2 ^{\frac{2}{3}}}{2}

w^{2} = 3 \frac{1}{16}

化简

3 \frac{1}{16} : \frac{2 ^{\frac{2}{3}}}{4}

3 \frac{1}{16}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3 1}{3 16}

316 = 232

316

16

质因数分解:

2^{4}

16

16

除以

216 = 8 \cdot 2

= 2 \cdot 8

8

除以

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2

= 2^{4}

= 3 2^{4}

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 3 2^{3} \cdot 2

使用根式运算法则:

n ab = n a n b

= 32 3 2^{3}

使用根式运算法则:

n a^{n} = a

3 2^{3} = 2

= 232

= \frac{3 1}{2 3 2}

使用法则

n 1 = 1

31 = 1

= \frac{1}{2 3 2}

\frac{1}{2 3 2}

有理化:

\frac{2 ^{\frac{2}{3}}}{4}

\frac{1}{2 3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= \frac{1 \cdot 2 ^{\frac{2}{3}}}{2 3 2 \cdot 2 ^{\frac{2}{3}}}

1 \cdot 2^{\frac{2}{3}} = 2^{\frac{2}{3}}

232 \cdot 2^{\frac{2}{3}} = 4

232 \cdot 2^{\frac{2}{3}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2 \cdot 2^{\frac{2}{3}} 32 = 2 \cdot 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{1 + \frac{2}{3} + \frac{1}{3}}

= 2^{1 + \frac{2}{3} + \frac{1}{3}}

化简

1 + \frac{2}{3} + \frac{1}{3} : 2

1 + \frac{2}{3} + \frac{1}{3}

将项转换为分式:

1 = \frac{1}{1}

= \frac{1}{1} + \frac{2}{3} + \frac{1}{3}

1, 3, 3

的最小公倍数:

3

1, 3, 3

最小公倍数 (LCM)

1

质因数分解

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

计算出由至少在以下一个数字中出现的因数组成的数字:

1, 3, 3

= 3

数字相乘:

3 = 3

= 3

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

3

对于

\frac{1}{1} :

将分母和分子乘以

3

\frac{1}{1} = \frac{1 \cdot 3}{1 \cdot 3} = \frac{3}{3}

= \frac{3}{3} + \frac{2}{3} + \frac{1}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 + 2 + 1}{3}

数字相加:

3 + 2 + 1 = 6

= \frac{6}{3}

数字相除:

\frac{6}{3} = 2

= 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 ^{\frac{2}{3}}}{4}

= \frac{2 ^{\frac{2}{3}}}{4}

w^{2} = \frac{2 ^{\frac{2}{3}}}{4}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

w = \frac{2 ^{\frac{2}{3}}}{4}, w = - \frac{2 ^{\frac{2}{3}}}{4}

\frac{2 ^{\frac{2}{3}}}{4} = \frac{2 ^{\frac{2}{3}}}{2}

\frac{2 ^{\frac{2}{3}}}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{2 ^{\frac{2}{3}}}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{2 ^{\frac{2}{3}}}{2}

- \frac{2 ^{\frac{2}{3}}}{4} = - \frac{2 ^{\frac{2}{3}}}{2}

- \frac{2 ^{\frac{2}{3}}}{4}

化简

\frac{2 ^{\frac{2}{3}}}{4} : \frac{2 ^{\frac{2}{3}}}{2}

\frac{2 ^{\frac{2}{3}}}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{2 ^{\frac{2}{3}}}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{2 ^{\frac{2}{3}}}{2}

= - \frac{2 ^{\frac{2}{3}}}{2}

w = \frac{2 ^{\frac{2}{3}}}{2}, w = - \frac{2 ^{\frac{2}{3}}}{2}

解

w^{2} = 3 \frac{1}{16} \frac{- 1 + 3 i}{2} : w = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, w = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

w^{2} = 3 \frac{1}{16} \frac{- 1 + 3 i}{2}

替代

w = u + v i

(u + v i)^{2} = 3 \frac{1}{16} \frac{- 1 + 3 i}{2}

乘开

(u + v i)^{2} = 3 \frac{1}{16} \frac{- 1 + 3 i}{2}

展开

(u + v i)^{2} : (u^{2} - v^{2}) + 2 i uv

(u + v i)^{2}

= (u + i v)^{2}

使用完全平方公式:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = u, b = v i

= u^{2} + 2 uv i + (v i)^{2}

(v i)^{2} = - v^{2}

(v i)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} v^{2}

i^{2} = - 1

i^{2}

使用虚数运算法则:

i^{2} = - 1

= - 1

= (- 1) v^{2}

整理后得

= - v^{2}

= u^{2} + 2 i uv - v^{2}

将

u^{2} + 2 i uv - v^{2}

改写成标准复数形式:

(u^{2} - v^{2}) + 2 uv i

u^{2} + 2 i uv - v^{2}

将复数的实部和虚部分组

= (u^{2} - v^{2}) + 2 uv i

= (u^{2} - v^{2}) + 2 uv i

展开

3 \frac{1}{16} \frac{- 1 + 3 i}{2} : - \frac{2 ^{\frac{2}{3}}}{8} + i \frac{2 ^{\frac{2}{3}} 3}{8}

3 \frac{1}{16} \frac{- 1 + 3 i}{2}

3 \frac{1}{16} = \frac{1}{2 3 2}

3 \frac{1}{16}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3 1}{3 16}

316 = 232

316

16

质因数分解:

2^{4}

16

16

除以

216 = 8 \cdot 2

= 2 \cdot 8

8

除以

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2

= 2^{4}

= 3 2^{4}

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 3 2^{3} \cdot 2

使用根式运算法则:

n ab = n a n b

= 32 3 2^{3}

使用根式运算法则:

n a^{n} = a

3 2^{3} = 2

= 232

= \frac{3 1}{2 3 2}

使用法则

n 1 = 1

31 = 1

= \frac{1}{2 3 2}

= \frac{1}{2 3 2} \cdot \frac{- 1 + 3 i}{2}

分式相乘:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot ( - 1 + 3 i )}{2 3 2 \cdot 2}

1 \cdot (- 1 + 3 i) = - 1 + 3 i

1 \cdot (- 1 + 3 i)

乘以:

1 \cdot (- 1 + 3 i) = (- 1 + 3 i)

= (- 1 + 3 i)

去除括号:

(- a) = - a

= - 1 + 3 i

= \frac{- 1 + 3 i}{2 \cdot 2 3 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 1 + 3 i}{4 3 2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 + 3 i}{4 3 2} = - \frac{1}{4 3 2} + \frac{3 i}{4 3 2}

= - \frac{1}{4 3 2} + \frac{3 i}{4 3 2}

将

- \frac{1}{4 3 2} + \frac{3 i}{4 3 2}

改写成标准复数形式:

- \frac{2 ^{\frac{2}{3}}}{8} + \frac{3 \cdot 2 ^{\frac{2}{3}}}{8} i

- \frac{1}{4 3 2} + \frac{3 i}{4 3 2}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 1 + 3 i}{4 3 2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 + 3 i}{4 3 2} = - \frac{1}{4 3 2} + \frac{3 i}{4 3 2}

= - \frac{1}{4 3 2} + \frac{3 i}{4 3 2}

\frac{3}{4 3 2} = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

\frac{3}{4 3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{4 3 2 \cdot 2 ^{\frac{2}{3}}}

432 \cdot 2^{\frac{2}{3}} = 8

432 \cdot 2^{\frac{2}{3}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 4 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1}

使用法则

a^{1} = a

= 2

= 4 \cdot 2

数字相乘:

4 \cdot 2 = 8

= 8

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

= - \frac{1}{4 3 2} + \frac{3 \cdot 2 ^{\frac{2}{3}}}{8} i

- \frac{1}{4 3 2} = - \frac{2 ^{\frac{2}{3}}}{8}

- \frac{1}{4 3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= - \frac{1 \cdot 2 ^{\frac{2}{3}}}{4 3 2 \cdot 2 ^{\frac{2}{3}}}

1 \cdot 2^{\frac{2}{3}} = 2^{\frac{2}{3}}

432 \cdot 2^{\frac{2}{3}} = 8

432 \cdot 2^{\frac{2}{3}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 4 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1}

使用法则

a^{1} = a

= 2

= 4 \cdot 2

数字相乘:

4 \cdot 2 = 8

= 8

= - \frac{2 ^{\frac{2}{3}}}{8}

= - \frac{2 ^{\frac{2}{3}}}{8} + \frac{3 \cdot 2 ^{\frac{2}{3}}}{8} i

= - \frac{2 ^{\frac{2}{3}}}{8} + \frac{3 \cdot 2 ^{\frac{2}{3}}}{8} i

(u^{2} - v^{2}) + 2 i uv = - \frac{2 ^{\frac{2}{3}}}{8} + i \frac{2 ^{\frac{2}{3}} 3}{8}

(u^{2} - v^{2}) + 2 i uv = - \frac{2 ^{\frac{2}{3}}}{8} + i \frac{2 ^{\frac{2}{3}} 3}{8}

复数仅在实部和虚部均相等时才相等改写为方程组:

[u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} 2 uv = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}]

[u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} 2 uv = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}] : (u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots v = - 0.54556 \dots)

[u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} 2 uv = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}]

对于

2 uv = \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

将

u

移到一边:

u = \frac{3}{2 ^{\frac{10}{3}} v}

2 uv = \frac{2 ^{\frac{2}{3}} 3}{8}

化简

\frac{2 ^{\frac{2}{3}} 3}{8} : \frac{3}{2 ^{\frac{7}{3}}}

\frac{2 ^{\frac{2}{3}} 3}{8}

因式分解数字:

8 = 2^{3}

= \frac{2 ^{\frac{2}{3}} 3}{2 ^{3}}

化简

\frac{2 ^{\frac{2}{3}}}{2 ^{3}} : \frac{1}{2 ^{\frac{7}{3}}}

\frac{2 ^{\frac{2}{3}}}{2 ^{3}}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

= \frac{1}{2 ^{3 - \frac{2}{3}}}

3 - \frac{2}{3} = \frac{7}{3}

3 - \frac{2}{3}

将项转换为分式:

3 = \frac{3 \cdot 3}{3}

= \frac{3 \cdot 3}{3} - \frac{2}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 \cdot 3 - 2}{3}

3 \cdot 3 - 2 = 7

3 \cdot 3 - 2

数字相乘:

3 \cdot 3 = 9

= 9 - 2

数字相减:

9 - 2 = 7

= 7

= \frac{7}{3}

= \frac{1}{2 ^{\frac{7}{3}}}

= \frac{3}{2 ^{\frac{7}{3}}}

2 uv = \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 v

2 uv = \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 v

\frac{2 uv}{2 v} = \frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 v}

化简

\frac{2 uv}{2 v} = \frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 v}

化简

\frac{2 uv}{2 v} : u

\frac{2 uv}{2 v}

约分:

2

= \frac{uv}{v}

约分:

v

= u

化简

\frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 v} : \frac{3}{2 ^{\frac{10}{3}} v}

\frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 v}

使用分式法则:

\frac{\frac{a}{b}}{c} = \frac{a}{b \cdot c}

= \frac{3}{2 ^{\frac{7}{3}} \cdot 2 v}

2^{\frac{7}{3}} \cdot 2 = 2^{\frac{10}{3}}

2^{\frac{7}{3}} \cdot 2

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{7}{3}} \cdot 2 = 2^{\frac{7}{3} + 1}

= 2^{\frac{7}{3} + 1}

\frac{7}{3} + 1 = \frac{10}{3}

\frac{7}{3} + 1

将项转换为分式:

1 = \frac{1 \cdot 3}{3}

= \frac{7}{3} + \frac{1 \cdot 3}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{7 + 1 \cdot 3}{3}

7 + 1 \cdot 3 = 10

7 + 1 \cdot 3

数字相乘:

1 \cdot 3 = 3

= 7 + 3

数字相加:

7 + 3 = 10

= 10

= \frac{10}{3}

= 2^{\frac{10}{3}}

= \frac{3}{2 ^{\frac{10}{3}} v}

u = \frac{3}{2 ^{\frac{10}{3}} v}

u = \frac{3}{2 ^{\frac{10}{3}} v}

u = \frac{3}{2 ^{\frac{10}{3}} v}

将解

u = \frac{3}{2 ^{\frac{10}{3}} v}

代入

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

对于

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

，用

\frac{3}{2 ^{\frac{10}{3}} v}

替代

u : v \approx 0.54556 \dots, v \approx - 0.54556 \dots

对于

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

，用

\frac{3}{2 ^{\frac{10}{3}} v}

替代

u

(\frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

解

(\frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} : v \approx 0.54556 \dots, v \approx - 0.54556 \dots

(\frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

乘以最小公倍数

(\frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

化简

(\frac{3}{2 ^{\frac{10}{3}} v})^{2} : \frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}}

(\frac{3}{2 ^{\frac{10}{3}} v})^{2}

\frac{3}{2 ^{\frac{10}{3}} v} = \frac{3}{2 ^{3} 3 2 v}

\frac{3}{2 ^{\frac{10}{3}} v}

2^{\frac{10}{3}} = 2^{3} 32

2^{\frac{10}{3}}

2^{\frac{10}{3}} = 2^{3 + \frac{1}{3}}

= 2^{3 + \frac{1}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{3} \cdot 2^{\frac{1}{3}}

整理后得

= 2^{3} 32

= \frac{3}{2 ^{3} 3 2 v}

= (\frac{3}{2 ^{3} 3 2 v})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{( 2 ^{3} 3 2 v ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(2^{3} 32 v)^{2} = (2^{3})^{2} (32)^{2} v^{2}

= \frac{( 3 ) ^{2}}{( 2 ^{3} ) ^{2} ( 3 2 ) ^{2} v ^{2}}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3}{( 2 ^{3} ) ^{2} ( 3 2 ) ^{2} v ^{2}}

(2^{3})^{2} : 2^{6}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 2^{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= 2^{6}

= \frac{3}{2 ^{6} ( 3 2 ) ^{2} v ^{2}}

(32)^{2} : 2^{\frac{2}{3}}

使用根式运算法则:

n a = a^{\frac{1}{n}}

= (2^{\frac{1}{3}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{3} \cdot 2}

\frac{1}{3} \cdot 2 = \frac{2}{3}

\frac{1}{3} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{3}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{3}

= 2^{\frac{2}{3}}

= \frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}}

2^{6} \cdot 2^{\frac{2}{3}} v^{2} = 2^{6 + \frac{2}{3}} v^{2}

2^{6} \cdot 2^{\frac{2}{3}} v^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{6} \cdot 2^{\frac{2}{3}} = 2^{6 + \frac{2}{3}}

= 2^{6 + \frac{2}{3}} v^{2}

= \frac{3}{2 ^{\frac{2}{3} + 6} v ^{2}}

2^{6 + \frac{2}{3}} = 2^{6} \cdot 2^{\frac{2}{3}}

2^{6 + \frac{2}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{6} \cdot 2^{\frac{2}{3}}

= \frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}}

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

找到

2^{6 + \frac{2}{3}} v^{2}, 8

的最小公倍数:

64 \cdot 2^{\frac{2}{3}} v^{2}

2^{6 + \frac{2}{3}} v^{2}, 8

最小公倍数 (LCM)

64, 8

的最小公倍数:

64

64, 8

最小公倍数 (LCM)

64

质因数分解:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

64

64

除以

264 = 32 \cdot 2

= 2 \cdot 32

32

除以

232 = 16 \cdot 2

= 2 \cdot 2 \cdot 16

16

除以

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 8

8

除以

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

8

质因数分解:

2 \cdot 2 \cdot 2

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

将每个因子乘以它在

64

或

8

中出现的最多次数

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

数字相乘:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64

= 64

计算出由出现在

64 \cdot 2^{\frac{2}{3}} v^{2}

或

8

中的因子组成的表达式

= 64 \cdot 2^{\frac{2}{3}} v^{2}

乘以最小公倍数=

64 \cdot 2^{\frac{2}{3}} v^{2}

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} - v^{2} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

化简

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} - v^{2} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

化简

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} : 3

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 64 \cdot 2 ^{\frac{2}{3}} v ^{2}}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}}

约分:

2^{\frac{2}{3}}

= \frac{3 \cdot 64 v ^{2}}{2 ^{6} v ^{2}}

约分:

v^{2}

= \frac{3 \cdot 64}{2 ^{6}}

数字相乘:

3 \cdot 64 = 192

= \frac{192}{2 ^{6}}

分解

192 : 2^{6} \cdot 3

因式分解

192 = 2^{6} \cdot 3

= \frac{2 ^{6} \cdot 3}{2 ^{6}}

约分:

2^{6}

= 3

化简

- v^{2} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} : - 64 \cdot 2^{\frac{2}{3}} v^{4}

- v^{2} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

v^{2} v^{2} = v^{2 + 2}

= - 64 \cdot 2^{\frac{2}{3}} v^{2 + 2}

数字相加:

2 + 2 = 4

= - 64 \cdot 2^{\frac{2}{3}} v^{4}

化简

- \frac{2 ^{\frac{2}{3}}}{8} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} : - 1632 v^{2}

- \frac{2 ^{\frac{2}{3}}}{8} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{2 ^{\frac{2}{3}} \cdot 64 \cdot 2 ^{\frac{2}{3}} v ^{2}}{8}

2^{\frac{2}{3}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} = 64 \cdot 2^{2 \cdot \frac{2}{3}} v^{2}

2^{\frac{2}{3}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} \cdot 2^{\frac{2}{3}} = 2^{\frac{2}{3} + \frac{2}{3}}

= 64 \cdot 2^{\frac{2}{3} + \frac{2}{3}} v^{2}

同类项相加:

\frac{2}{3} + \frac{2}{3} = 2 \cdot \frac{2}{3}

= 64 \cdot 2^{2 \cdot \frac{2}{3}} v^{2}

= - \frac{64 \cdot 2 ^{2 \cdot \frac{2}{3}} v ^{2}}{8}

数字相除:

\frac{64}{8} = 8

= - 8 \cdot 2^{2 \cdot \frac{2}{3}} v^{2}

2^{2 \cdot \frac{2}{3}} = 232

2^{2 \cdot \frac{2}{3}}

乘

2 \cdot \frac{2}{3} : \frac{4}{3}

2 \cdot \frac{2}{3}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 \cdot 2}{3}

数字相乘:

2 \cdot 2 = 4

= \frac{4}{3}

= 2^{\frac{4}{3}}

2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}}

= 2^{1 + \frac{1}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{3}}

整理后得

= 232

= - 8 \cdot 232 v^{2}

数字相乘:

8 \cdot 2 = 16

= - 1632 v^{2}

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

解

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2} : v \approx 0.54556 \dots, v \approx - 0.54556 \dots

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

将

1632 v^{2}

para o lado esquerdo

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

两边加上

1632 v^{2}

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} + 1632 v^{2} = - 1632 v^{2} + 1632 v^{2}

化简

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} + 1632 v^{2} = 0

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} + 1632 v^{2} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 64 \cdot 2^{\frac{2}{3}} v^{4} + 1632 v^{2} + 3 = 0

使用牛顿-拉弗森方法找到

- 101.59366 \dots v^{4} + 20.15873 \dots v^{2} + 3 = 0

的一个解:

v \approx 0.54556 \dots

- 101.59366 \dots v^{4} + 20.15873 \dots v^{2} + 3 = 0

牛顿-拉弗森近似法定义

f (v) = - 101.59366 \dots v^{4} + 20.15873 \dots v^{2} + 3

找到

f^{^{'}} (v) : - 406.37466 \dots v^{3} + 40.31747 \dots v

\frac{d}{d v} (- 101.59366 \dots v^{4} + 20.15873 \dots v^{2} + 3)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d v} (101.59366 \dots v^{4}) + \frac{d}{d v} (20.15873 \dots v^{2}) + \frac{d}{d v} (3)

\frac{d}{d v} (101.59366 \dots v^{4}) = 406.37466 \dots v^{3}

\frac{d}{d v} (101.59366 \dots v^{4})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 101.59366 \dots \frac{d}{d v} (v^{4})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 101.59366 \dots \cdot 4 v^{4 - 1}

化简

= 406.37466 \dots v^{3}

\frac{d}{d v} (20.15873 \dots v^{2}) = 40.31747 \dots v

\frac{d}{d v} (20.15873 \dots v^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 20.15873 \dots \frac{d}{d v} (v^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 20.15873 \dots \cdot 2 v^{2 - 1}

化简

= 40.31747 \dots v

\frac{d}{d v} (3) = 0

\frac{d}{d v} (3)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= - 406.37466 \dots v^{3} + 40.31747 \dots v + 0

化简

= - 406.37466 \dots v^{3} + 40.31747 \dots v

令

v_{0} = 1

计算

v_{n + 1}

至

Δ v_{n + 1} < 0.000001

v_{1} = 0.78573 \dots : Δ v_{1} = 0.21426 \dots

f (v_{0}) = - 101.59366 \dots \cdot 1^{4} + 20.15873 \dots \cdot 1^{2} + 3 = - 78.43493 \dots

f^{^{'}} (v_{0}) = - 406.37466 \dots \cdot 1^{3} + 40.31747 \dots \cdot 1 = - 366.05719 \dots

v_{1} = 0.78573 \dots

Δ v_{1} = ∣ 0.78573 \dots - 1 ∣ = 0.21426 \dots

Δ v_{1} = 0.21426 \dots

v_{2} = 0.64504 \dots : Δ v_{2} = 0.14068 \dots

f (v_{1}) = - 101.59366 \dots \cdot 0.78573 \dots^{4} + 20.15873 \dots \cdot 0.78573 \dots^{2} + 3 = - 23.27682 \dots

f^{^{'}} (v_{1}) = - 406.37466 \dots \cdot 0.78573 \dots^{3} + 40.31747 \dots \cdot 0.78573 \dots = - 165.44887 \dots

v_{2} = 0.64504 \dots

Δ v_{2} = ∣ 0.64504 \dots - 0.78573 \dots ∣ = 0.14068 \dots

Δ v_{2} = 0.14068 \dots

v_{3} = 0.57039 \dots : Δ v_{3} = 0.07465 \dots

f (v_{2}) = - 101.59366 \dots \cdot 0.64504 \dots^{4} + 20.15873 \dots \cdot 0.64504 \dots^{2} + 3 = - 6.20040 \dots

f^{^{'}} (v_{2}) = - 406.37466 \dots \cdot 0.64504 \dots^{3} + 40.31747 \dots \cdot 0.64504 \dots = - 83.05958 \dots

v_{3} = 0.57039 \dots

Δ v_{3} = ∣ 0.57039 \dots - 0.64504 \dots ∣ = 0.07465 \dots

Δ v_{3} = 0.07465 \dots

v_{4} = 0.54759 \dots : Δ v_{4} = 0.02280 \dots

f (v_{3}) = - 101.59366 \dots \cdot 0.57039 \dots^{4} + 20.15873 \dots \cdot 0.57039 \dots^{2} + 3 = - 1.19513 \dots

f^{^{'}} (v_{3}) = - 406.37466 \dots \cdot 0.57039 \dots^{3} + 40.31747 \dots \cdot 0.57039 \dots = - 52.41610 \dots

v_{4} = 0.54759 \dots

Δ v_{4} = ∣ 0.54759 \dots - 0.57039 \dots ∣ = 0.02280 \dots

Δ v_{4} = 0.02280 \dots

v_{5} = 0.54557 \dots : Δ v_{5} = 0.00201 \dots

f (v_{4}) = - 101.59366 \dots \cdot 0.54759 \dots^{4} + 20.15873 \dots \cdot 0.54759 \dots^{2} + 3 = - 0.08990 \dots

f^{^{'}} (v_{4}) = - 406.37466 \dots \cdot 0.54759 \dots^{3} + 40.31747 \dots \cdot 0.54759 \dots = - 44.64836 \dots

v_{5} = 0.54557 \dots

Δ v_{5} = ∣ 0.54557 \dots - 0.54759 \dots ∣ = 0.00201 \dots

Δ v_{5} = 0.00201 \dots

v_{6} = 0.54556 \dots : Δ v_{6} = 0.00001 \dots

f (v_{5}) = - 101.59366 \dots \cdot 0.54557 \dots^{4} + 20.15873 \dots \cdot 0.54557 \dots^{2} + 3 = - 0.00065 \dots

f^{^{'}} (v_{5}) = - 406.37466 \dots \cdot 0.54557 \dots^{3} + 40.31747 \dots \cdot 0.54557 \dots = - 43.99616 \dots

v_{6} = 0.54556 \dots

Δ v_{6} = ∣ 0.54556 \dots - 0.54557 \dots ∣ = 0.00001 \dots

Δ v_{6} = 0.00001 \dots

v_{7} = 0.54556 \dots : Δ v_{7} = 8.18838 E - 10

f (v_{6}) = - 101.59366 \dots \cdot 0.54556 \dots^{4} + 20.15873 \dots \cdot 0.54556 \dots^{2} + 3 = - 3.60218 E - 8

f^{^{'}} (v_{6}) = - 406.37466 \dots \cdot 0.54556 \dots^{3} + 40.31747 \dots \cdot 0.54556 \dots = - 43.99134 \dots

v_{7} = 0.54556 \dots

Δ v_{7} = ∣ 0.54556 \dots - 0.54556 \dots ∣ = 8.18838 E - 10

Δ v_{7} = 8.18838 E - 10

v \approx 0.54556 \dots

使用长除法

Eq u a t i o n 0 : \frac{- 64 \cdot 2 ^{\frac{2}{3}} v ^{4} + 16 3 2 v ^{2} + 3}{v - 0.54556 \dots} = - 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots

- 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots \approx 0

使用牛顿-拉弗森方法找到

- 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots = 0

的一个解:

v \approx - 0.54556 \dots

- 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots = 0

牛顿-拉弗森近似法定义

f (v) = - 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots

找到

f^{^{'}} (v) : - 304.78100 \dots v^{2} - 110.85125 \dots v - 10.07936 \dots

\frac{d}{d v} (- 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d v} (101.59366 \dots v^{3}) - \frac{d}{d v} (55.42562 \dots v^{2}) - \frac{d}{d v} (10.07936 \dots v) - \frac{d}{d v} (5.49891 \dots)

\frac{d}{d v} (101.59366 \dots v^{3}) = 304.78100 \dots v^{2}

\frac{d}{d v} (101.59366 \dots v^{3})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 101.59366 \dots \frac{d}{d v} (v^{3})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 101.59366 \dots \cdot 3 v^{3 - 1}

化简

= 304.78100 \dots v^{2}

\frac{d}{d v} (55.42562 \dots v^{2}) = 110.85125 \dots v

\frac{d}{d v} (55.42562 \dots v^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 55.42562 \dots \frac{d}{d v} (v^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 55.42562 \dots \cdot 2 v^{2 - 1}

化简

= 110.85125 \dots v

\frac{d}{d v} (10.07936 \dots v) = 10.07936 \dots

\frac{d}{d v} (10.07936 \dots v)

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 10.07936 \dots \frac{d v}{d v}

使用常见微分定则:

\frac{d v}{d v} = 1

= 10.07936 \dots \cdot 1

化简

= 10.07936 \dots

\frac{d}{d v} (5.49891 \dots) = 0

\frac{d}{d v} (5.49891 \dots)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= - 304.78100 \dots v^{2} - 110.85125 \dots v - 10.07936 \dots - 0

化简

= - 304.78100 \dots v^{2} - 110.85125 \dots v - 10.07936 \dots

令

v_{0} = - 1

计算

v_{n + 1}

至

Δ v_{n + 1} < 0.000001

v_{1} = - 0.75124 \dots : Δ v_{1} = 0.24875 \dots

f (v_{0}) = - 101.59366 \dots (- 1)^{3} - 55.42562 \dots (- 1)^{2} - 10.07936 \dots (- 1) - 5.49891 \dots = 50.74849 \dots

f^{^{'}} (v_{0}) = - 304.78100 \dots (- 1)^{2} - 110.85125 \dots (- 1) - 10.07936 \dots = - 204.00911 \dots

v_{1} = - 0.75124 \dots

Δ v_{1} = ∣ - 0.75124 \dots - (- 1) ∣ = 0.24875 \dots

Δ v_{1} = 0.24875 \dots

v_{2} = - 0.61091 \dots : Δ v_{2} = 0.14032 \dots

f (v_{1}) = - 101.59366 \dots (- 0.75124 \dots)^{3} - 55.42562 \dots (- 0.75124 \dots)^{2} - 10.07936 \dots (- 0.75124 \dots) - 5.49891 \dots = 13.86617 \dots

f^{^{'}} (v_{1}) = - 304.78100 \dots (- 0.75124 \dots)^{2} - 110.85125 \dots (- 0.75124 \dots) - 10.07936 \dots = - 98.81153 \dots

v_{2} = - 0.61091 \dots

Δ v_{2} = ∣ - 0.61091 \dots - (- 0.75124 \dots) ∣ = 0.14032 \dots

Δ v_{2} = 0.14032 \dots

v_{3} = - 0.55501 \dots : Δ v_{3} = 0.05590 \dots

f (v_{2}) = - 101.59366 \dots (- 0.61091 \dots)^{3} - 55.42562 \dots (- 0.61091 \dots)^{2} - 10.07936 \dots (- 0.61091 \dots) - 5.49891 \dots = 3.13665 \dots

f^{^{'}} (v_{2}) = - 304.78100 \dots (- 0.61091 \dots)^{2} - 110.85125 \dots (- 0.61091 \dots) - 10.07936 \dots = - 56.10803 \dots

v_{3} = - 0.55501 \dots

Δ v_{3} = ∣ - 0.55501 \dots - (- 0.61091 \dots) ∣ = 0.05590 \dots

Δ v_{3} = 0.05590 \dots

v_{4} = - 0.54579 \dots : Δ v_{4} = 0.00921 \dots

f (v_{3}) = - 101.59366 \dots (- 0.55501 \dots)^{3} - 55.42562 \dots (- 0.55501 \dots)^{2} - 10.07936 \dots (- 0.55501 \dots) - 5.49891 \dots = 0.39093 \dots

f^{^{'}} (v_{3}) = - 304.78100 \dots (- 0.55501 \dots)^{2} - 110.85125 \dots (- 0.55501 \dots) - 10.07936 \dots = - 42.43951 \dots

v_{4} = - 0.54579 \dots

Δ v_{4} = ∣ - 0.54579 \dots - (- 0.55501 \dots) ∣ = 0.00921 \dots

Δ v_{4} = 0.00921 \dots

v_{5} = - 0.54556 \dots : Δ v_{5} = 0.00023 \dots

f (v_{4}) = - 101.59366 \dots (- 0.54579 \dots)^{3} - 55.42562 \dots (- 0.54579 \dots)^{2} - 10.07936 \dots (- 0.54579 \dots) - 5.49891 \dots = 0.00957 \dots

f^{^{'}} (v_{4}) = - 304.78100 \dots (- 0.54579 \dots)^{2} - 110.85125 \dots (- 0.54579 \dots) - 10.07936 \dots = - 40.37008 \dots

v_{5} = - 0.54556 \dots

Δ v_{5} = ∣ - 0.54556 \dots - (- 0.54579 \dots) ∣ = 0.00023 \dots

Δ v_{5} = 0.00023 \dots

v_{6} = - 0.54556 \dots : Δ v_{6} = 1.54611 E - 7

f (v_{5}) = - 101.59366 \dots (- 0.54556 \dots)^{3} - 55.42562 \dots (- 0.54556 \dots)^{2} - 10.07936 \dots (- 0.54556 \dots) - 5.49891 \dots = 6.23352 E - 6

f^{^{'}} (v_{5}) = - 304.78100 \dots (- 0.54556 \dots)^{2} - 110.85125 \dots (- 0.54556 \dots) - 10.07936 \dots = - 40.31750 \dots

v_{6} = - 0.54556 \dots

Δ v_{6} = ∣ - 0.54556 \dots - (- 0.54556 \dots) ∣ = 1.54611 E - 7

Δ v_{6} = 1.54611 E - 7

v \approx - 0.54556 \dots

使用长除法

Eq u a t i o n 0 : \frac{- 101.59366 \dots v ^{3} - 55.42562 \dots v ^{2} - 10.07936 \dots v - 5.49891 \dots}{v + 0.54556 \dots} = - 101.59366 \dots v^{2} - 10.07936 \dots

- 101.59366 \dots v^{2} - 10.07936 \dots \approx 0

使用牛顿-拉弗森方法找到

- 101.59366 \dots v^{2} - 10.07936 \dots = 0

的一个解:

v \in R

无解

- 101.59366 \dots v^{2} - 10.07936 \dots = 0

牛顿-拉弗森近似法定义

f (v) = - 101.59366 \dots v^{2} - 10.07936 \dots

找到

f^{^{'}} (v) : - 203.18733 \dots v

\frac{d}{d v} (- 101.59366 \dots v^{2} - 10.07936 \dots)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d v} (101.59366 \dots v^{2}) - \frac{d}{d v} (10.07936 \dots)

\frac{d}{d v} (101.59366 \dots v^{2}) = 203.18733 \dots v

\frac{d}{d v} (101.59366 \dots v^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 101.59366 \dots \frac{d}{d v} (v^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 101.59366 \dots \cdot 2 v^{2 - 1}

化简

= 203.18733 \dots v

\frac{d}{d v} (10.07936 \dots) = 0

\frac{d}{d v} (10.07936 \dots)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= - 203.18733 \dots v - 0

化简

= - 203.18733 \dots v

令

v_{0} = - 1

计算

v_{n + 1}

至

Δ v_{n + 1} < 0.000001

v_{1} = - 0.45039 \dots : Δ v_{1} = 0.54960 \dots

f (v_{0}) = - 101.59366 \dots (- 1)^{2} - 10.07936 \dots = - 111.67303 \dots

f^{^{'}} (v_{0}) = - 203.18733 \dots (- 1) = 203.18733 \dots

v_{1} = - 0.45039 \dots

Δ v_{1} = ∣ - 0.45039 \dots - (- 1) ∣ = 0.54960 \dots

Δ v_{1} = 0.54960 \dots

v_{2} = - 0.11505 \dots : Δ v_{2} = 0.33533 \dots

f (v_{1}) = - 101.59366 \dots (- 0.45039 \dots)^{2} - 10.07936 \dots = - 30.68810 \dots

f^{^{'}} (v_{1}) = - 203.18733 \dots (- 0.45039 \dots) = 91.51429 \dots

v_{2} = - 0.11505 \dots

Δ v_{2} = ∣ - 0.11505 \dots - (- 0.45039 \dots) ∣ = 0.33533 \dots

Δ v_{2} = 0.33533 \dots

v_{3} = 0.37361 \dots : Δ v_{3} = 0.48867 \dots

f (v_{2}) = - 101.59366 \dots (- 0.11505 \dots)^{2} - 10.07936 \dots = - 11.42427 \dots

f^{^{'}} (v_{2}) = - 203.18733 \dots (- 0.11505 \dots) = 23.37813 \dots

v_{3} = 0.37361 \dots

Δ v_{3} = ∣ 0.37361 \dots - (- 0.11505 \dots) ∣ = 0.48867 \dots

Δ v_{3} = 0.48867 \dots

v_{4} = 0.05403 \dots : Δ v_{4} = 0.31958 \dots

f (v_{3}) = - 101.59366 \dots \cdot 0.37361 \dots^{2} - 10.07936 \dots = - 24.26076 \dots

f^{^{'}} (v_{3}) = - 203.18733 \dots \cdot 0.37361 \dots = - 75.91416 \dots

v_{4} = 0.05403 \dots

Δ v_{4} = ∣ 0.05403 \dots - 0.37361 \dots ∣ = 0.31958 \dots

Δ v_{4} = 0.31958 \dots

v_{5} = - 0.89102 \dots : Δ v_{5} = 0.94505 \dots

f (v_{4}) = - 101.59366 \dots \cdot 0.05403 \dots^{2} - 10.07936 \dots = - 10.37600 \dots

f^{^{'}} (v_{4}) = - 203.18733 \dots \cdot 0.05403 \dots = - 10.97924 \dots

v_{5} = - 0.89102 \dots

Δ v_{5} = ∣ - 0.89102 \dots - 0.05403 \dots ∣ = 0.94505 \dots

Δ v_{5} = 0.94505 \dots

v_{6} = - 0.38983 \dots : Δ v_{6} = 0.50118 \dots

f (v_{5}) = - 101.59366 \dots (- 0.89102 \dots)^{2} - 10.07936 \dots = - 90.73642 \dots

f^{^{'}} (v_{5}) = - 203.18733 \dots (- 0.89102 \dots) = 181.04414 \dots

v_{6} = - 0.38983 \dots

Δ v_{6} = ∣ - 0.38983 \dots - (- 0.89102 \dots) ∣ = 0.50118 \dots

Δ v_{6} = 0.50118 \dots

v_{7} = - 0.06766 \dots : Δ v_{7} = 0.32216 \dots

f (v_{6}) = - 101.59366 \dots (- 0.38983 \dots)^{2} - 10.07936 \dots = - 25.51884 \dots

f^{^{'}} (v_{6}) = - 203.18733 \dots (- 0.38983 \dots) = 79.20991 \dots

v_{7} = - 0.06766 \dots

Δ v_{7} = ∣ - 0.06766 \dots - (- 0.38983 \dots) ∣ = 0.32216 \dots

Δ v_{7} = 0.32216 \dots

v_{8} = 0.69923 \dots : Δ v_{8} = 0.76690 \dots

f (v_{7}) = - 101.59366 \dots (- 0.06766 \dots)^{2} - 10.07936 \dots = - 10.54458 \dots

f^{^{'}} (v_{7}) = - 203.18733 \dots (- 0.06766 \dots) = 13.74960 \dots

v_{8} = 0.69923 \dots

Δ v_{8} = ∣ 0.69923 \dots - (- 0.06766 \dots) ∣ = 0.76690 \dots

Δ v_{8} = 0.76690 \dots

v_{9} = 0.27867 \dots : Δ v_{9} = 0.42055 \dots

f (v_{8}) = - 101.59366 \dots \cdot 0.69923 \dots^{2} - 10.07936 \dots = - 59.75094 \dots

f^{^{'}} (v_{8}) = - 203.18733 \dots \cdot 0.69923 \dots = - 142.07487 \dots

v_{9} = 0.27867 \dots

Δ v_{9} = ∣ 0.27867 \dots - 0.69923 \dots ∣ = 0.42055 \dots

Δ v_{9} = 0.42055 \dots

v_{10} = - 0.03867 \dots : Δ v_{10} = 0.31734 \dots

f (v_{9}) = - 101.59366 \dots \cdot 0.27867 \dots^{2} - 10.07936 \dots = - 17.96890 \dots

f^{^{'}} (v_{9}) = - 203.18733 \dots \cdot 0.27867 \dots = - 56.62250 \dots

v_{10} = - 0.03867 \dots

Δ v_{10} = ∣ - 0.03867 \dots - 0.27867 \dots ∣ = 0.31734 \dots

Δ v_{10} = 0.31734 \dots

v_{11} = 1.26333 \dots : Δ v_{11} = 1.30200 \dots

f (v_{10}) = - 101.59366 \dots (- 0.03867 \dots)^{2} - 10.07936 \dots = - 10.23132 \dots

f^{^{'}} (v_{10}) = - 203.18733 \dots (- 0.03867 \dots) = 7.85811 \dots

v_{11} = 1.26333 \dots

Δ v_{11} = ∣ 1.26333 \dots - (- 0.03867 \dots) ∣ = 1.30200 \dots

Δ v_{11} = 1.30200 \dots

无法得出解

解为

v \approx 0.54556 \dots, v \approx - 0.54556 \dots

v \approx 0.54556 \dots, v \approx - 0.54556 \dots

验证解

找到无定义的点（奇点）:

v = 0

取

(\frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2}

的分母，令其等于零

解

2^{\frac{10}{3}} v = 0 : v = 0

2^{\frac{10}{3}} v = 0

两边除以

2^{\frac{10}{3}}

2^{\frac{10}{3}} v = 0

两边除以

2^{\frac{10}{3}}

\frac{2 ^{\frac{10}{3}} v}{2 ^{\frac{10}{3}}} = \frac{0}{2 ^{\frac{10}{3}}}

化简

\frac{2 ^{\frac{10}{3}} v}{2 ^{\frac{10}{3}}} = \frac{0}{2 ^{\frac{10}{3}}}

化简

\frac{2 ^{\frac{10}{3}} v}{2 ^{\frac{10}{3}}} : v

\frac{2 ^{\frac{10}{3}} v}{2 ^{\frac{10}{3}}}

约分:

2^{\frac{10}{3}}

= v

化简

\frac{0}{2 ^{\frac{10}{3}}} : 0

\frac{0}{2 ^{\frac{10}{3}}}

使用法则

\frac{0}{a} = 0

:

a \neq = 0

= 0

v = 0

v = 0

v = 0

以下点无定义

v = 0

将不在定义域的点与解相综合:

v \approx 0.54556 \dots, v \approx - 0.54556 \dots

将解

v = 0.54556 \dots, v = - 0.54556 \dots

代入

2 uv = \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

对于

2 uv = \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

，用

0.54556 \dots

替代

v : u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

对于

2 uv = \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

，用

0.54556 \dots

替代

v

2 u \cdot 0.54556 \dots = \frac{2 ^{\frac{2}{3}} 3}{8}

解

2 u 0.54556 \dots = \frac{2 ^{\frac{2}{3}} 3}{8} : u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

2 u \cdot 0.54556 \dots = \frac{2 ^{\frac{2}{3}} 3}{8}

化简

\frac{2 ^{\frac{2}{3}} 3}{8} : \frac{3}{2 ^{\frac{7}{3}}}

\frac{2 ^{\frac{2}{3}} 3}{8}

因式分解数字:

8 = 2^{3}

= \frac{2 ^{\frac{2}{3}} 3}{2 ^{3}}

化简

\frac{2 ^{\frac{2}{3}}}{2 ^{3}} : \frac{1}{2 ^{\frac{7}{3}}}

\frac{2 ^{\frac{2}{3}}}{2 ^{3}}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

= \frac{1}{2 ^{3 - \frac{2}{3}}}

3 - \frac{2}{3} = \frac{7}{3}

3 - \frac{2}{3}

将项转换为分式:

3 = \frac{3 \cdot 3}{3}

= \frac{3 \cdot 3}{3} - \frac{2}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 \cdot 3 - 2}{3}

3 \cdot 3 - 2 = 7

3 \cdot 3 - 2

数字相乘:

3 \cdot 3 = 9

= 9 - 2

数字相减:

9 - 2 = 7

= 7

= \frac{7}{3}

= \frac{1}{2 ^{\frac{7}{3}}}

= \frac{3}{2 ^{\frac{7}{3}}}

2 u \cdot 0.54556 \dots = \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 \cdot 0.54556 \dots

2 u \cdot 0.54556 \dots = \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 \cdot 0.54556 \dots

\frac{2 u \cdot 0.54556 \dots}{2 \cdot 0.54556 \dots} = \frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 \cdot 0.54556 \dots}

化简

\frac{2 u \cdot 0.54556 \dots}{2 \cdot 0.54556 \dots} = \frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 \cdot 0.54556 \dots}

化简

\frac{2 u \cdot 0.54556 \dots}{2 \cdot 0.54556 \dots} : u

\frac{2 u \cdot 0.54556 \dots}{2 \cdot 0.54556 \dots}

约分:

2

= \frac{u \cdot 0.54556 \dots}{0.54556 \dots}

约分:

0.54556 \dots

= u

化简

\frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 \cdot 0.54556 \dots} : \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

\frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 \cdot 0.54556 \dots}

数字相乘:

2 \cdot 0.54556 \dots = 1.09112 \dots

= \frac{\frac{3}{2 ^{\frac{7}{3}}}}{1.09112 \dots}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3}{2 ^{\frac{7}{3}} \cdot 1.09112 \dots}

2^{\frac{7}{3}} = 2^{2} 32

2^{\frac{7}{3}}

2^{\frac{7}{3}} = 2^{2 + \frac{1}{3}}

= 2^{2 + \frac{1}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{2} \cdot 2^{\frac{1}{3}}

整理后得

= 2^{2} 32

= \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

对于

2 uv = \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

，用

- 0.54556 \dots

替代

v : u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

对于

2 uv = \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

，用

- 0.54556 \dots

替代

v

2 u (- 0.54556 \dots) = \frac{2 ^{\frac{2}{3}} 3}{8}

解

2 u (- 0.54556 \dots) = \frac{2 ^{\frac{2}{3}} 3}{8} : u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

2 u (- 0.54556 \dots) = \frac{2 ^{\frac{2}{3}} 3}{8}

化简

\frac{2 ^{\frac{2}{3}} 3}{8} : \frac{3}{2 ^{\frac{7}{3}}}

\frac{2 ^{\frac{2}{3}} 3}{8}

因式分解数字:

8 = 2^{3}

= \frac{2 ^{\frac{2}{3}} 3}{2 ^{3}}

化简

\frac{2 ^{\frac{2}{3}}}{2 ^{3}} : \frac{1}{2 ^{\frac{7}{3}}}

\frac{2 ^{\frac{2}{3}}}{2 ^{3}}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

= \frac{1}{2 ^{3 - \frac{2}{3}}}

3 - \frac{2}{3} = \frac{7}{3}

3 - \frac{2}{3}

将项转换为分式:

3 = \frac{3 \cdot 3}{3}

= \frac{3 \cdot 3}{3} - \frac{2}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 \cdot 3 - 2}{3}

3 \cdot 3 - 2 = 7

3 \cdot 3 - 2

数字相乘:

3 \cdot 3 = 9

= 9 - 2

数字相减:

9 - 2 = 7

= 7

= \frac{7}{3}

= \frac{1}{2 ^{\frac{7}{3}}}

= \frac{3}{2 ^{\frac{7}{3}}}

2 u (- 0.54556 \dots) = \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 (- 0.54556 \dots)

2 u (- 0.54556 \dots) = \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 (- 0.54556 \dots)

\frac{2 u ( - 0.54556 \dots )}{2 ( - 0.54556 \dots )} = \frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 ( - 0.54556 \dots )}

化简

\frac{2 u ( - 0.54556 \dots )}{2 ( - 0.54556 \dots )} = \frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 ( - 0.54556 \dots )}

化简

\frac{2 u ( - 0.54556 \dots )}{2 ( - 0.54556 \dots )} : u

\frac{2 u ( - 0.54556 \dots )}{2 ( - 0.54556 \dots )}

数字相乘:

2 (- 0.54556 \dots) = - 1.09112 \dots

= \frac{- 1.09112 \dots u}{- 1.09112 \dots}

约分:

- 1.09112 \dots

= u

化简

\frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 ( - 0.54556 \dots )} : - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

\frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 ( - 0.54556 \dots )}

去除括号:

(- a) = - a

= \frac{\frac{3}{2 ^{\frac{7}{3}}}}{- 2 \cdot 0.54556 \dots}

数字相乘:

2 \cdot 0.54556 \dots = 1.09112 \dots

= \frac{\frac{3}{2 ^{\frac{7}{3}}}}{- 1.09112 \dots}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{\frac{3}{2 ^{\frac{7}{3}}}}{1.09112 \dots}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

\frac{\frac{3}{2 ^{\frac{7}{3}}}}{1.09112 \dots} = \frac{3}{2 ^{\frac{7}{3}} \cdot 1.09112 \dots}

= - \frac{3}{2 ^{\frac{7}{3}} \cdot 1.09112 \dots}

2^{\frac{7}{3}} = 2^{2} 32

2^{\frac{7}{3}}

2^{\frac{7}{3}} = 2^{2 + \frac{1}{3}}

= 2^{2 + \frac{1}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{2} \cdot 2^{\frac{1}{3}}

整理后得

= 2^{2} 32

= - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

将解代入原方程进行验证

将它们代入

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

检验解是否符合

去除与方程不符的解。

检验

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = - 0.54556 \dots

的解:

真

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

代入

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = - 0.54556 \dots

(- \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots})^{2} - (- 0.54556 \dots)^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

整理后得

\frac{3 3 2}{38.09762 \dots} - 0.29763 \dots = - \frac{2 ^{\frac{2}{3}}}{8}

真

检验

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots

的解:

真

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

代入

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots

(\frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots})^{2} - 0.54556 \dots^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

整理后得

\frac{3 3 2}{38.09762 \dots} - 0.29763 \dots = - \frac{2 ^{\frac{2}{3}}}{8}

真

将它们代入

2 uv = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

检验解是否符合

去除与方程不符的解。

检验

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = - 0.54556 \dots

的解:

真

2 uv = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

代入

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = - 0.54556 \dots

2 (- \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}) (- 0.54556 \dots) = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

整理后得

\frac{3 \cdot 2 ^{\frac{2}{3}} \cdot 0.54556 \dots}{4.36449 \dots} = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

真

检验

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots

的解:

真

2 uv = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

代入

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots

2 \cdot \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} \cdot 0.54556 \dots = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

整理后得

\frac{3 \cdot 2 ^{\frac{2}{3}} \cdot 0.54556 \dots}{4.36449 \dots} = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

真

因而，

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}, 2 uv = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

最后的解是

(u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots v = - 0.54556 \dots)

w = u + v i

代回

w = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, w = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

解

w^{2} = 3 \frac{1}{16} \frac{- 1 - 3 i}{2} : w = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, w = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

w^{2} = 3 \frac{1}{16} \frac{- 1 - 3 i}{2}

替代

w = u + v i

(u + v i)^{2} = 3 \frac{1}{16} \frac{- 1 - 3 i}{2}

乘开

(u + v i)^{2} = 3 \frac{1}{16} \frac{- 1 - 3 i}{2}

展开

(u + v i)^{2} : (u^{2} - v^{2}) + 2 i uv

(u + v i)^{2}

= (u + i v)^{2}

使用完全平方公式:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = u, b = v i

= u^{2} + 2 uv i + (v i)^{2}

(v i)^{2} = - v^{2}

(v i)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} v^{2}

i^{2} = - 1

i^{2}

使用虚数运算法则:

i^{2} = - 1

= - 1

= (- 1) v^{2}

整理后得

= - v^{2}

= u^{2} + 2 i uv - v^{2}

将

u^{2} + 2 i uv - v^{2}

改写成标准复数形式:

(u^{2} - v^{2}) + 2 uv i

u^{2} + 2 i uv - v^{2}

将复数的实部和虚部分组

= (u^{2} - v^{2}) + 2 uv i

= (u^{2} - v^{2}) + 2 uv i

展开

3 \frac{1}{16} \frac{- 1 - 3 i}{2} : - \frac{2 ^{\frac{2}{3}}}{8} - i \frac{2 ^{\frac{2}{3}} 3}{8}

3 \frac{1}{16} \frac{- 1 - 3 i}{2}

3 \frac{1}{16} = \frac{1}{2 3 2}

3 \frac{1}{16}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3 1}{3 16}

316 = 232

316

16

质因数分解:

2^{4}

16

16

除以

216 = 8 \cdot 2

= 2 \cdot 8

8

除以

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2

= 2^{4}

= 3 2^{4}

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 3 2^{3} \cdot 2

使用根式运算法则:

n ab = n a n b

= 32 3 2^{3}

使用根式运算法则:

n a^{n} = a

3 2^{3} = 2

= 232

= \frac{3 1}{2 3 2}

使用法则

n 1 = 1

31 = 1

= \frac{1}{2 3 2}

= \frac{1}{2 3 2} \cdot \frac{- 1 - 3 i}{2}

分式相乘:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot ( - 1 - 3 i )}{2 3 2 \cdot 2}

1 \cdot (- 1 - 3 i) = - 1 - 3 i

1 \cdot (- 1 - 3 i)

乘以:

1 \cdot (- 1 - 3 i) = (- 1 - 3 i)

= (- 1 - 3 i)

去除括号:

(- a) = - a

= - 1 - 3 i

= \frac{- 1 - 3 i}{2 \cdot 2 3 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 1 - 3 i}{4 3 2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 - 3 i}{4 3 2} = - \frac{1}{4 3 2} - \frac{3 i}{4 3 2}

= - \frac{1}{4 3 2} - \frac{3 i}{4 3 2}

将

- \frac{1}{4 3 2} - \frac{3 i}{4 3 2}

改写成标准复数形式:

- \frac{2 ^{\frac{2}{3}}}{8} - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8} i

- \frac{1}{4 3 2} - \frac{3 i}{4 3 2}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 1 - 3 i}{4 3 2}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{- 1 - 3 i}{4 3 2} = - \frac{1}{4 3 2} - \frac{3 i}{4 3 2}

= - \frac{1}{4 3 2} - \frac{3 i}{4 3 2}

- \frac{3}{4 3 2} = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

- \frac{3}{4 3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= - \frac{3 \cdot 2 ^{\frac{2}{3}}}{4 3 2 \cdot 2 ^{\frac{2}{3}}}

432 \cdot 2^{\frac{2}{3}} = 8

432 \cdot 2^{\frac{2}{3}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 4 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1}

使用法则

a^{1} = a

= 2

= 4 \cdot 2

数字相乘:

4 \cdot 2 = 8

= 8

= - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

= - \frac{1}{4 3 2} - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8} i

- \frac{1}{4 3 2} = - \frac{2 ^{\frac{2}{3}}}{8}

- \frac{1}{4 3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= - \frac{1 \cdot 2 ^{\frac{2}{3}}}{4 3 2 \cdot 2 ^{\frac{2}{3}}}

1 \cdot 2^{\frac{2}{3}} = 2^{\frac{2}{3}}

432 \cdot 2^{\frac{2}{3}} = 8

432 \cdot 2^{\frac{2}{3}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 4 \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1}

使用法则

a^{1} = a

= 2

= 4 \cdot 2

数字相乘:

4 \cdot 2 = 8

= 8

= - \frac{2 ^{\frac{2}{3}}}{8}

= - \frac{2 ^{\frac{2}{3}}}{8} - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8} i

= - \frac{2 ^{\frac{2}{3}}}{8} - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8} i

(u^{2} - v^{2}) + 2 i uv = - \frac{2 ^{\frac{2}{3}}}{8} - i \frac{2 ^{\frac{2}{3}} 3}{8}

(u^{2} - v^{2}) + 2 i uv = - \frac{2 ^{\frac{2}{3}}}{8} - i \frac{2 ^{\frac{2}{3}} 3}{8}

复数仅在实部和虚部均相等时才相等改写为方程组:

[u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} 2 uv = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}]

[u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} 2 uv = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}] : (u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots v = - 0.54556 \dots)

[u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} 2 uv = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}]

对于

2 uv = - \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

将

u

移到一边:

u = - \frac{3}{2 ^{\frac{10}{3}} v}

2 uv = - \frac{2 ^{\frac{2}{3}} 3}{8}

化简

- \frac{2 ^{\frac{2}{3}} 3}{8} : - \frac{3}{2 ^{\frac{7}{3}}}

- \frac{2 ^{\frac{2}{3}} 3}{8}

因式分解数字:

8 = 2^{3}

= - \frac{2 ^{\frac{2}{3}} 3}{2 ^{3}}

化简

\frac{2 ^{\frac{2}{3}}}{2 ^{3}} : \frac{1}{2 ^{\frac{7}{3}}}

\frac{2 ^{\frac{2}{3}}}{2 ^{3}}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

= \frac{1}{2 ^{3 - \frac{2}{3}}}

3 - \frac{2}{3} = \frac{7}{3}

3 - \frac{2}{3}

将项转换为分式:

3 = \frac{3 \cdot 3}{3}

= \frac{3 \cdot 3}{3} - \frac{2}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 \cdot 3 - 2}{3}

3 \cdot 3 - 2 = 7

3 \cdot 3 - 2

数字相乘:

3 \cdot 3 = 9

= 9 - 2

数字相减:

9 - 2 = 7

= 7

= \frac{7}{3}

= \frac{1}{2 ^{\frac{7}{3}}}

= - \frac{3}{2 ^{\frac{7}{3}}}

2 uv = - \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 v

2 uv = - \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 v

\frac{2 uv}{2 v} = \frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 v}

化简

\frac{2 uv}{2 v} = \frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 v}

化简

\frac{2 uv}{2 v} : u

\frac{2 uv}{2 v}

约分:

2

= \frac{uv}{v}

约分:

v

= u

化简

\frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 v} : - \frac{3}{2 ^{\frac{10}{3}} v}

\frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 v}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 v}

使用分式法则:

\frac{\frac{a}{b}}{c} = \frac{a}{b \cdot c}

\frac{\frac{3}{2 ^{\frac{7}{3}}}}{2 v} = \frac{3}{2 ^{\frac{7}{3}} \cdot 2 v}

= - \frac{3}{2 ^{\frac{7}{3}} \cdot 2 v}

2^{\frac{7}{3}} \cdot 2 = 2^{\frac{10}{3}}

2^{\frac{7}{3}} \cdot 2

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{7}{3}} \cdot 2 = 2^{\frac{7}{3} + 1}

= 2^{\frac{7}{3} + 1}

\frac{7}{3} + 1 = \frac{10}{3}

\frac{7}{3} + 1

将项转换为分式:

1 = \frac{1 \cdot 3}{3}

= \frac{7}{3} + \frac{1 \cdot 3}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{7 + 1 \cdot 3}{3}

7 + 1 \cdot 3 = 10

7 + 1 \cdot 3

数字相乘:

1 \cdot 3 = 3

= 7 + 3

数字相加:

7 + 3 = 10

= 10

= \frac{10}{3}

= 2^{\frac{10}{3}}

= - \frac{3}{2 ^{\frac{10}{3}} v}

u = - \frac{3}{2 ^{\frac{10}{3}} v}

u = - \frac{3}{2 ^{\frac{10}{3}} v}

u = - \frac{3}{2 ^{\frac{10}{3}} v}

将解

u = - \frac{3}{2 ^{\frac{10}{3}} v}

代入

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

对于

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

，用

- \frac{3}{2 ^{\frac{10}{3}} v}

替代

u : v \approx 0.54556 \dots, v \approx - 0.54556 \dots

对于

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

，用

- \frac{3}{2 ^{\frac{10}{3}} v}

替代

u

(- \frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

解

(- \frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} : v \approx 0.54556 \dots, v \approx - 0.54556 \dots

(- \frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

乘以最小公倍数

(- \frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

化简

(- \frac{3}{2 ^{\frac{10}{3}} v})^{2} : \frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}}

(- \frac{3}{2 ^{\frac{10}{3}} v})^{2}

\frac{3}{2 ^{\frac{10}{3}} v} = \frac{3}{2 ^{3} 3 2 v}

\frac{3}{2 ^{\frac{10}{3}} v}

2^{\frac{10}{3}} = 2^{3} 32

2^{\frac{10}{3}}

2^{\frac{10}{3}} = 2^{3 + \frac{1}{3}}

= 2^{3 + \frac{1}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{3} \cdot 2^{\frac{1}{3}}

整理后得

= 2^{3} 32

= \frac{3}{2 ^{3} 3 2 v}

= (- \frac{3}{2 ^{3} 3 2 v})^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{3}{2 ^{3} 3 2 v})^{2} = (\frac{3}{2 ^{3} 3 2 v})^{2}

= (\frac{3}{2 ^{3} 3 2 v})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 3 ) ^{2}}{( 2 ^{3} 3 2 v ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(2^{3} 32 v)^{2} = (2^{3})^{2} (32)^{2} v^{2}

= \frac{( 3 ) ^{2}}{( 2 ^{3} ) ^{2} ( 3 2 ) ^{2} v ^{2}}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= \frac{3}{( 2 ^{3} ) ^{2} ( 3 2 ) ^{2} v ^{2}}

(2^{3})^{2} : 2^{6}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 2^{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= 2^{6}

= \frac{3}{2 ^{6} ( 3 2 ) ^{2} v ^{2}}

(32)^{2} : 2^{\frac{2}{3}}

使用根式运算法则:

n a = a^{\frac{1}{n}}

= (2^{\frac{1}{3}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{3} \cdot 2}

\frac{1}{3} \cdot 2 = \frac{2}{3}

\frac{1}{3} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{3}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{3}

= 2^{\frac{2}{3}}

= \frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}}

2^{6} \cdot 2^{\frac{2}{3}} v^{2} = 2^{6 + \frac{2}{3}} v^{2}

2^{6} \cdot 2^{\frac{2}{3}} v^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{6} \cdot 2^{\frac{2}{3}} = 2^{6 + \frac{2}{3}}

= 2^{6 + \frac{2}{3}} v^{2}

= \frac{3}{2 ^{\frac{2}{3} + 6} v ^{2}}

2^{6 + \frac{2}{3}} = 2^{6} \cdot 2^{\frac{2}{3}}

2^{6 + \frac{2}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{6} \cdot 2^{\frac{2}{3}}

= \frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}}

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

找到

2^{6 + \frac{2}{3}} v^{2}, 8

的最小公倍数:

64 \cdot 2^{\frac{2}{3}} v^{2}

2^{6 + \frac{2}{3}} v^{2}, 8

最小公倍数 (LCM)

64, 8

的最小公倍数:

64

64, 8

最小公倍数 (LCM)

64

质因数分解:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

64

64

除以

264 = 32 \cdot 2

= 2 \cdot 32

32

除以

232 = 16 \cdot 2

= 2 \cdot 2 \cdot 16

16

除以

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 8

8

除以

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

8

质因数分解:

2 \cdot 2 \cdot 2

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

将每个因子乘以它在

64

或

8

中出现的最多次数

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

数字相乘:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64

= 64

计算出由出现在

64 \cdot 2^{\frac{2}{3}} v^{2}

或

8

中的因子组成的表达式

= 64 \cdot 2^{\frac{2}{3}} v^{2}

乘以最小公倍数=

64 \cdot 2^{\frac{2}{3}} v^{2}

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} - v^{2} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

化简

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} - v^{2} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} = - \frac{2 ^{\frac{2}{3}}}{8} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

化简

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} : 3

\frac{3}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 \cdot 64 \cdot 2 ^{\frac{2}{3}} v ^{2}}{2 ^{6} \cdot 2 ^{\frac{2}{3}} v ^{2}}

约分:

2^{\frac{2}{3}}

= \frac{3 \cdot 64 v ^{2}}{2 ^{6} v ^{2}}

约分:

v^{2}

= \frac{3 \cdot 64}{2 ^{6}}

数字相乘:

3 \cdot 64 = 192

= \frac{192}{2 ^{6}}

分解

192 : 2^{6} \cdot 3

因式分解

192 = 2^{6} \cdot 3

= \frac{2 ^{6} \cdot 3}{2 ^{6}}

约分:

2^{6}

= 3

化简

- v^{2} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} : - 64 \cdot 2^{\frac{2}{3}} v^{4}

- v^{2} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

v^{2} v^{2} = v^{2 + 2}

= - 64 \cdot 2^{\frac{2}{3}} v^{2 + 2}

数字相加:

2 + 2 = 4

= - 64 \cdot 2^{\frac{2}{3}} v^{4}

化简

- \frac{2 ^{\frac{2}{3}}}{8} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} : - 1632 v^{2}

- \frac{2 ^{\frac{2}{3}}}{8} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{2 ^{\frac{2}{3}} \cdot 64 \cdot 2 ^{\frac{2}{3}} v ^{2}}{8}

2^{\frac{2}{3}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2} = 64 \cdot 2^{2 \cdot \frac{2}{3}} v^{2}

2^{\frac{2}{3}} \cdot 64 \cdot 2^{\frac{2}{3}} v^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} \cdot 2^{\frac{2}{3}} = 2^{\frac{2}{3} + \frac{2}{3}}

= 64 \cdot 2^{\frac{2}{3} + \frac{2}{3}} v^{2}

同类项相加:

\frac{2}{3} + \frac{2}{3} = 2 \cdot \frac{2}{3}

= 64 \cdot 2^{2 \cdot \frac{2}{3}} v^{2}

= - \frac{64 \cdot 2 ^{2 \cdot \frac{2}{3}} v ^{2}}{8}

数字相除:

\frac{64}{8} = 8

= - 8 \cdot 2^{2 \cdot \frac{2}{3}} v^{2}

2^{2 \cdot \frac{2}{3}} = 232

2^{2 \cdot \frac{2}{3}}

乘

2 \cdot \frac{2}{3} : \frac{4}{3}

2 \cdot \frac{2}{3}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 \cdot 2}{3}

数字相乘:

2 \cdot 2 = 4

= \frac{4}{3}

= 2^{\frac{4}{3}}

2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}}

= 2^{1 + \frac{1}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{3}}

整理后得

= 232

= - 8 \cdot 232 v^{2}

数字相乘:

8 \cdot 2 = 16

= - 1632 v^{2}

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

解

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2} : v \approx 0.54556 \dots, v \approx - 0.54556 \dots

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

将

1632 v^{2}

para o lado esquerdo

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} = - 1632 v^{2}

两边加上

1632 v^{2}

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} + 1632 v^{2} = - 1632 v^{2} + 1632 v^{2}

化简

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} + 1632 v^{2} = 0

3 - 64 \cdot 2^{\frac{2}{3}} v^{4} + 1632 v^{2} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 64 \cdot 2^{\frac{2}{3}} v^{4} + 1632 v^{2} + 3 = 0

使用牛顿-拉弗森方法找到

- 101.59366 \dots v^{4} + 20.15873 \dots v^{2} + 3 = 0

的一个解:

v \approx 0.54556 \dots

- 101.59366 \dots v^{4} + 20.15873 \dots v^{2} + 3 = 0

牛顿-拉弗森近似法定义

f (v) = - 101.59366 \dots v^{4} + 20.15873 \dots v^{2} + 3

找到

f^{^{'}} (v) : - 406.37466 \dots v^{3} + 40.31747 \dots v

\frac{d}{d v} (- 101.59366 \dots v^{4} + 20.15873 \dots v^{2} + 3)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d v} (101.59366 \dots v^{4}) + \frac{d}{d v} (20.15873 \dots v^{2}) + \frac{d}{d v} (3)

\frac{d}{d v} (101.59366 \dots v^{4}) = 406.37466 \dots v^{3}

\frac{d}{d v} (101.59366 \dots v^{4})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 101.59366 \dots \frac{d}{d v} (v^{4})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 101.59366 \dots \cdot 4 v^{4 - 1}

化简

= 406.37466 \dots v^{3}

\frac{d}{d v} (20.15873 \dots v^{2}) = 40.31747 \dots v

\frac{d}{d v} (20.15873 \dots v^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 20.15873 \dots \frac{d}{d v} (v^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 20.15873 \dots \cdot 2 v^{2 - 1}

化简

= 40.31747 \dots v

\frac{d}{d v} (3) = 0

\frac{d}{d v} (3)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= - 406.37466 \dots v^{3} + 40.31747 \dots v + 0

化简

= - 406.37466 \dots v^{3} + 40.31747 \dots v

令

v_{0} = 1

计算

v_{n + 1}

至

Δ v_{n + 1} < 0.000001

v_{1} = 0.78573 \dots : Δ v_{1} = 0.21426 \dots

f (v_{0}) = - 101.59366 \dots \cdot 1^{4} + 20.15873 \dots \cdot 1^{2} + 3 = - 78.43493 \dots

f^{^{'}} (v_{0}) = - 406.37466 \dots \cdot 1^{3} + 40.31747 \dots \cdot 1 = - 366.05719 \dots

v_{1} = 0.78573 \dots

Δ v_{1} = ∣ 0.78573 \dots - 1 ∣ = 0.21426 \dots

Δ v_{1} = 0.21426 \dots

v_{2} = 0.64504 \dots : Δ v_{2} = 0.14068 \dots

f (v_{1}) = - 101.59366 \dots \cdot 0.78573 \dots^{4} + 20.15873 \dots \cdot 0.78573 \dots^{2} + 3 = - 23.27682 \dots

f^{^{'}} (v_{1}) = - 406.37466 \dots \cdot 0.78573 \dots^{3} + 40.31747 \dots \cdot 0.78573 \dots = - 165.44887 \dots

v_{2} = 0.64504 \dots

Δ v_{2} = ∣ 0.64504 \dots - 0.78573 \dots ∣ = 0.14068 \dots

Δ v_{2} = 0.14068 \dots

v_{3} = 0.57039 \dots : Δ v_{3} = 0.07465 \dots

f (v_{2}) = - 101.59366 \dots \cdot 0.64504 \dots^{4} + 20.15873 \dots \cdot 0.64504 \dots^{2} + 3 = - 6.20040 \dots

f^{^{'}} (v_{2}) = - 406.37466 \dots \cdot 0.64504 \dots^{3} + 40.31747 \dots \cdot 0.64504 \dots = - 83.05958 \dots

v_{3} = 0.57039 \dots

Δ v_{3} = ∣ 0.57039 \dots - 0.64504 \dots ∣ = 0.07465 \dots

Δ v_{3} = 0.07465 \dots

v_{4} = 0.54759 \dots : Δ v_{4} = 0.02280 \dots

f (v_{3}) = - 101.59366 \dots \cdot 0.57039 \dots^{4} + 20.15873 \dots \cdot 0.57039 \dots^{2} + 3 = - 1.19513 \dots

f^{^{'}} (v_{3}) = - 406.37466 \dots \cdot 0.57039 \dots^{3} + 40.31747 \dots \cdot 0.57039 \dots = - 52.41610 \dots

v_{4} = 0.54759 \dots

Δ v_{4} = ∣ 0.54759 \dots - 0.57039 \dots ∣ = 0.02280 \dots

Δ v_{4} = 0.02280 \dots

v_{5} = 0.54557 \dots : Δ v_{5} = 0.00201 \dots

f (v_{4}) = - 101.59366 \dots \cdot 0.54759 \dots^{4} + 20.15873 \dots \cdot 0.54759 \dots^{2} + 3 = - 0.08990 \dots

f^{^{'}} (v_{4}) = - 406.37466 \dots \cdot 0.54759 \dots^{3} + 40.31747 \dots \cdot 0.54759 \dots = - 44.64836 \dots

v_{5} = 0.54557 \dots

Δ v_{5} = ∣ 0.54557 \dots - 0.54759 \dots ∣ = 0.00201 \dots

Δ v_{5} = 0.00201 \dots

v_{6} = 0.54556 \dots : Δ v_{6} = 0.00001 \dots

f (v_{5}) = - 101.59366 \dots \cdot 0.54557 \dots^{4} + 20.15873 \dots \cdot 0.54557 \dots^{2} + 3 = - 0.00065 \dots

f^{^{'}} (v_{5}) = - 406.37466 \dots \cdot 0.54557 \dots^{3} + 40.31747 \dots \cdot 0.54557 \dots = - 43.99616 \dots

v_{6} = 0.54556 \dots

Δ v_{6} = ∣ 0.54556 \dots - 0.54557 \dots ∣ = 0.00001 \dots

Δ v_{6} = 0.00001 \dots

v_{7} = 0.54556 \dots : Δ v_{7} = 8.18838 E - 10

f (v_{6}) = - 101.59366 \dots \cdot 0.54556 \dots^{4} + 20.15873 \dots \cdot 0.54556 \dots^{2} + 3 = - 3.60218 E - 8

f^{^{'}} (v_{6}) = - 406.37466 \dots \cdot 0.54556 \dots^{3} + 40.31747 \dots \cdot 0.54556 \dots = - 43.99134 \dots

v_{7} = 0.54556 \dots

Δ v_{7} = ∣ 0.54556 \dots - 0.54556 \dots ∣ = 8.18838 E - 10

Δ v_{7} = 8.18838 E - 10

v \approx 0.54556 \dots

使用长除法

Eq u a t i o n 0 : \frac{- 64 \cdot 2 ^{\frac{2}{3}} v ^{4} + 16 3 2 v ^{2} + 3}{v - 0.54556 \dots} = - 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots

- 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots \approx 0

使用牛顿-拉弗森方法找到

- 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots = 0

的一个解:

v \approx - 0.54556 \dots

- 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots = 0

牛顿-拉弗森近似法定义

f (v) = - 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots

找到

f^{^{'}} (v) : - 304.78100 \dots v^{2} - 110.85125 \dots v - 10.07936 \dots

\frac{d}{d v} (- 101.59366 \dots v^{3} - 55.42562 \dots v^{2} - 10.07936 \dots v - 5.49891 \dots)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d v} (101.59366 \dots v^{3}) - \frac{d}{d v} (55.42562 \dots v^{2}) - \frac{d}{d v} (10.07936 \dots v) - \frac{d}{d v} (5.49891 \dots)

\frac{d}{d v} (101.59366 \dots v^{3}) = 304.78100 \dots v^{2}

\frac{d}{d v} (101.59366 \dots v^{3})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 101.59366 \dots \frac{d}{d v} (v^{3})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 101.59366 \dots \cdot 3 v^{3 - 1}

化简

= 304.78100 \dots v^{2}

\frac{d}{d v} (55.42562 \dots v^{2}) = 110.85125 \dots v

\frac{d}{d v} (55.42562 \dots v^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 55.42562 \dots \frac{d}{d v} (v^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 55.42562 \dots \cdot 2 v^{2 - 1}

化简

= 110.85125 \dots v

\frac{d}{d v} (10.07936 \dots v) = 10.07936 \dots

\frac{d}{d v} (10.07936 \dots v)

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 10.07936 \dots \frac{d v}{d v}

使用常见微分定则:

\frac{d v}{d v} = 1

= 10.07936 \dots \cdot 1

化简

= 10.07936 \dots

\frac{d}{d v} (5.49891 \dots) = 0

\frac{d}{d v} (5.49891 \dots)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= - 304.78100 \dots v^{2} - 110.85125 \dots v - 10.07936 \dots - 0

化简

= - 304.78100 \dots v^{2} - 110.85125 \dots v - 10.07936 \dots

令

v_{0} = - 1

计算

v_{n + 1}

至

Δ v_{n + 1} < 0.000001

v_{1} = - 0.75124 \dots : Δ v_{1} = 0.24875 \dots

f (v_{0}) = - 101.59366 \dots (- 1)^{3} - 55.42562 \dots (- 1)^{2} - 10.07936 \dots (- 1) - 5.49891 \dots = 50.74849 \dots

f^{^{'}} (v_{0}) = - 304.78100 \dots (- 1)^{2} - 110.85125 \dots (- 1) - 10.07936 \dots = - 204.00911 \dots

v_{1} = - 0.75124 \dots

Δ v_{1} = ∣ - 0.75124 \dots - (- 1) ∣ = 0.24875 \dots

Δ v_{1} = 0.24875 \dots

v_{2} = - 0.61091 \dots : Δ v_{2} = 0.14032 \dots

f (v_{1}) = - 101.59366 \dots (- 0.75124 \dots)^{3} - 55.42562 \dots (- 0.75124 \dots)^{2} - 10.07936 \dots (- 0.75124 \dots) - 5.49891 \dots = 13.86617 \dots

f^{^{'}} (v_{1}) = - 304.78100 \dots (- 0.75124 \dots)^{2} - 110.85125 \dots (- 0.75124 \dots) - 10.07936 \dots = - 98.81153 \dots

v_{2} = - 0.61091 \dots

Δ v_{2} = ∣ - 0.61091 \dots - (- 0.75124 \dots) ∣ = 0.14032 \dots

Δ v_{2} = 0.14032 \dots

v_{3} = - 0.55501 \dots : Δ v_{3} = 0.05590 \dots

f (v_{2}) = - 101.59366 \dots (- 0.61091 \dots)^{3} - 55.42562 \dots (- 0.61091 \dots)^{2} - 10.07936 \dots (- 0.61091 \dots) - 5.49891 \dots = 3.13665 \dots

f^{^{'}} (v_{2}) = - 304.78100 \dots (- 0.61091 \dots)^{2} - 110.85125 \dots (- 0.61091 \dots) - 10.07936 \dots = - 56.10803 \dots

v_{3} = - 0.55501 \dots

Δ v_{3} = ∣ - 0.55501 \dots - (- 0.61091 \dots) ∣ = 0.05590 \dots

Δ v_{3} = 0.05590 \dots

v_{4} = - 0.54579 \dots : Δ v_{4} = 0.00921 \dots

f (v_{3}) = - 101.59366 \dots (- 0.55501 \dots)^{3} - 55.42562 \dots (- 0.55501 \dots)^{2} - 10.07936 \dots (- 0.55501 \dots) - 5.49891 \dots = 0.39093 \dots

f^{^{'}} (v_{3}) = - 304.78100 \dots (- 0.55501 \dots)^{2} - 110.85125 \dots (- 0.55501 \dots) - 10.07936 \dots = - 42.43951 \dots

v_{4} = - 0.54579 \dots

Δ v_{4} = ∣ - 0.54579 \dots - (- 0.55501 \dots) ∣ = 0.00921 \dots

Δ v_{4} = 0.00921 \dots

v_{5} = - 0.54556 \dots : Δ v_{5} = 0.00023 \dots

f (v_{4}) = - 101.59366 \dots (- 0.54579 \dots)^{3} - 55.42562 \dots (- 0.54579 \dots)^{2} - 10.07936 \dots (- 0.54579 \dots) - 5.49891 \dots = 0.00957 \dots

f^{^{'}} (v_{4}) = - 304.78100 \dots (- 0.54579 \dots)^{2} - 110.85125 \dots (- 0.54579 \dots) - 10.07936 \dots = - 40.37008 \dots

v_{5} = - 0.54556 \dots

Δ v_{5} = ∣ - 0.54556 \dots - (- 0.54579 \dots) ∣ = 0.00023 \dots

Δ v_{5} = 0.00023 \dots

v_{6} = - 0.54556 \dots : Δ v_{6} = 1.54611 E - 7

f (v_{5}) = - 101.59366 \dots (- 0.54556 \dots)^{3} - 55.42562 \dots (- 0.54556 \dots)^{2} - 10.07936 \dots (- 0.54556 \dots) - 5.49891 \dots = 6.23352 E - 6

f^{^{'}} (v_{5}) = - 304.78100 \dots (- 0.54556 \dots)^{2} - 110.85125 \dots (- 0.54556 \dots) - 10.07936 \dots = - 40.31750 \dots

v_{6} = - 0.54556 \dots

Δ v_{6} = ∣ - 0.54556 \dots - (- 0.54556 \dots) ∣ = 1.54611 E - 7

Δ v_{6} = 1.54611 E - 7

v \approx - 0.54556 \dots

使用长除法

Eq u a t i o n 0 : \frac{- 101.59366 \dots v ^{3} - 55.42562 \dots v ^{2} - 10.07936 \dots v - 5.49891 \dots}{v + 0.54556 \dots} = - 101.59366 \dots v^{2} - 10.07936 \dots

- 101.59366 \dots v^{2} - 10.07936 \dots \approx 0

使用牛顿-拉弗森方法找到

- 101.59366 \dots v^{2} - 10.07936 \dots = 0

的一个解:

v \in R

无解

- 101.59366 \dots v^{2} - 10.07936 \dots = 0

牛顿-拉弗森近似法定义

f (v) = - 101.59366 \dots v^{2} - 10.07936 \dots

找到

f^{^{'}} (v) : - 203.18733 \dots v

\frac{d}{d v} (- 101.59366 \dots v^{2} - 10.07936 \dots)

使用微分加减法定则:

(f \pm g)^{'} = f^{'} \pm g^{'}

= - \frac{d}{d v} (101.59366 \dots v^{2}) - \frac{d}{d v} (10.07936 \dots)

\frac{d}{d v} (101.59366 \dots v^{2}) = 203.18733 \dots v

\frac{d}{d v} (101.59366 \dots v^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 101.59366 \dots \frac{d}{d v} (v^{2})

使用幂法则:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 101.59366 \dots \cdot 2 v^{2 - 1}

化简

= 203.18733 \dots v

\frac{d}{d v} (10.07936 \dots) = 0

\frac{d}{d v} (10.07936 \dots)

常数微分:

\frac{d}{d x} (a) = 0

= 0

= - 203.18733 \dots v - 0

化简

= - 203.18733 \dots v

令

v_{0} = - 1

计算

v_{n + 1}

至

Δ v_{n + 1} < 0.000001

v_{1} = - 0.45039 \dots : Δ v_{1} = 0.54960 \dots

f (v_{0}) = - 101.59366 \dots (- 1)^{2} - 10.07936 \dots = - 111.67303 \dots

f^{^{'}} (v_{0}) = - 203.18733 \dots (- 1) = 203.18733 \dots

v_{1} = - 0.45039 \dots

Δ v_{1} = ∣ - 0.45039 \dots - (- 1) ∣ = 0.54960 \dots

Δ v_{1} = 0.54960 \dots

v_{2} = - 0.11505 \dots : Δ v_{2} = 0.33533 \dots

f (v_{1}) = - 101.59366 \dots (- 0.45039 \dots)^{2} - 10.07936 \dots = - 30.68810 \dots

f^{^{'}} (v_{1}) = - 203.18733 \dots (- 0.45039 \dots) = 91.51429 \dots

v_{2} = - 0.11505 \dots

Δ v_{2} = ∣ - 0.11505 \dots - (- 0.45039 \dots) ∣ = 0.33533 \dots

Δ v_{2} = 0.33533 \dots

v_{3} = 0.37361 \dots : Δ v_{3} = 0.48867 \dots

f (v_{2}) = - 101.59366 \dots (- 0.11505 \dots)^{2} - 10.07936 \dots = - 11.42427 \dots

f^{^{'}} (v_{2}) = - 203.18733 \dots (- 0.11505 \dots) = 23.37813 \dots

v_{3} = 0.37361 \dots

Δ v_{3} = ∣ 0.37361 \dots - (- 0.11505 \dots) ∣ = 0.48867 \dots

Δ v_{3} = 0.48867 \dots

v_{4} = 0.05403 \dots : Δ v_{4} = 0.31958 \dots

f (v_{3}) = - 101.59366 \dots \cdot 0.37361 \dots^{2} - 10.07936 \dots = - 24.26076 \dots

f^{^{'}} (v_{3}) = - 203.18733 \dots \cdot 0.37361 \dots = - 75.91416 \dots

v_{4} = 0.05403 \dots

Δ v_{4} = ∣ 0.05403 \dots - 0.37361 \dots ∣ = 0.31958 \dots

Δ v_{4} = 0.31958 \dots

v_{5} = - 0.89102 \dots : Δ v_{5} = 0.94505 \dots

f (v_{4}) = - 101.59366 \dots \cdot 0.05403 \dots^{2} - 10.07936 \dots = - 10.37600 \dots

f^{^{'}} (v_{4}) = - 203.18733 \dots \cdot 0.05403 \dots = - 10.97924 \dots

v_{5} = - 0.89102 \dots

Δ v_{5} = ∣ - 0.89102 \dots - 0.05403 \dots ∣ = 0.94505 \dots

Δ v_{5} = 0.94505 \dots

v_{6} = - 0.38983 \dots : Δ v_{6} = 0.50118 \dots

f (v_{5}) = - 101.59366 \dots (- 0.89102 \dots)^{2} - 10.07936 \dots = - 90.73642 \dots

f^{^{'}} (v_{5}) = - 203.18733 \dots (- 0.89102 \dots) = 181.04414 \dots

v_{6} = - 0.38983 \dots

Δ v_{6} = ∣ - 0.38983 \dots - (- 0.89102 \dots) ∣ = 0.50118 \dots

Δ v_{6} = 0.50118 \dots

v_{7} = - 0.06766 \dots : Δ v_{7} = 0.32216 \dots

f (v_{6}) = - 101.59366 \dots (- 0.38983 \dots)^{2} - 10.07936 \dots = - 25.51884 \dots

f^{^{'}} (v_{6}) = - 203.18733 \dots (- 0.38983 \dots) = 79.20991 \dots

v_{7} = - 0.06766 \dots

Δ v_{7} = ∣ - 0.06766 \dots - (- 0.38983 \dots) ∣ = 0.32216 \dots

Δ v_{7} = 0.32216 \dots

v_{8} = 0.69923 \dots : Δ v_{8} = 0.76690 \dots

f (v_{7}) = - 101.59366 \dots (- 0.06766 \dots)^{2} - 10.07936 \dots = - 10.54458 \dots

f^{^{'}} (v_{7}) = - 203.18733 \dots (- 0.06766 \dots) = 13.74960 \dots

v_{8} = 0.69923 \dots

Δ v_{8} = ∣ 0.69923 \dots - (- 0.06766 \dots) ∣ = 0.76690 \dots

Δ v_{8} = 0.76690 \dots

v_{9} = 0.27867 \dots : Δ v_{9} = 0.42055 \dots

f (v_{8}) = - 101.59366 \dots \cdot 0.69923 \dots^{2} - 10.07936 \dots = - 59.75094 \dots

f^{^{'}} (v_{8}) = - 203.18733 \dots \cdot 0.69923 \dots = - 142.07487 \dots

v_{9} = 0.27867 \dots

Δ v_{9} = ∣ 0.27867 \dots - 0.69923 \dots ∣ = 0.42055 \dots

Δ v_{9} = 0.42055 \dots

v_{10} = - 0.03867 \dots : Δ v_{10} = 0.31734 \dots

f (v_{9}) = - 101.59366 \dots \cdot 0.27867 \dots^{2} - 10.07936 \dots = - 17.96890 \dots

f^{^{'}} (v_{9}) = - 203.18733 \dots \cdot 0.27867 \dots = - 56.62250 \dots

v_{10} = - 0.03867 \dots

Δ v_{10} = ∣ - 0.03867 \dots - 0.27867 \dots ∣ = 0.31734 \dots

Δ v_{10} = 0.31734 \dots

v_{11} = 1.26333 \dots : Δ v_{11} = 1.30200 \dots

f (v_{10}) = - 101.59366 \dots (- 0.03867 \dots)^{2} - 10.07936 \dots = - 10.23132 \dots

f^{^{'}} (v_{10}) = - 203.18733 \dots (- 0.03867 \dots) = 7.85811 \dots

v_{11} = 1.26333 \dots

Δ v_{11} = ∣ 1.26333 \dots - (- 0.03867 \dots) ∣ = 1.30200 \dots

Δ v_{11} = 1.30200 \dots

无法得出解

解为

v \approx 0.54556 \dots, v \approx - 0.54556 \dots

v \approx 0.54556 \dots, v \approx - 0.54556 \dots

验证解

找到无定义的点（奇点）:

v = 0

取

(- \frac{3}{2 ^{\frac{10}{3}} v})^{2} - v^{2}

的分母，令其等于零

解

2^{\frac{10}{3}} v = 0 : v = 0

2^{\frac{10}{3}} v = 0

两边除以

2^{\frac{10}{3}}

2^{\frac{10}{3}} v = 0

两边除以

2^{\frac{10}{3}}

\frac{2 ^{\frac{10}{3}} v}{2 ^{\frac{10}{3}}} = \frac{0}{2 ^{\frac{10}{3}}}

化简

\frac{2 ^{\frac{10}{3}} v}{2 ^{\frac{10}{3}}} = \frac{0}{2 ^{\frac{10}{3}}}

化简

\frac{2 ^{\frac{10}{3}} v}{2 ^{\frac{10}{3}}} : v

\frac{2 ^{\frac{10}{3}} v}{2 ^{\frac{10}{3}}}

约分:

2^{\frac{10}{3}}

= v

化简

\frac{0}{2 ^{\frac{10}{3}}} : 0

\frac{0}{2 ^{\frac{10}{3}}}

使用法则

\frac{0}{a} = 0

:

a \neq = 0

= 0

v = 0

v = 0

v = 0

以下点无定义

v = 0

将不在定义域的点与解相综合:

v \approx 0.54556 \dots, v \approx - 0.54556 \dots

将解

v = 0.54556 \dots, v = - 0.54556 \dots

代入

2 uv = - \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

对于

2 uv = - \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

，用

0.54556 \dots

替代

v : u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

对于

2 uv = - \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

，用

0.54556 \dots

替代

v

2 u \cdot 0.54556 \dots = - \frac{2 ^{\frac{2}{3}} 3}{8}

解

2 u 0.54556 \dots = - \frac{2 ^{\frac{2}{3}} 3}{8} : u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

2 u \cdot 0.54556 \dots = - \frac{2 ^{\frac{2}{3}} 3}{8}

化简

- \frac{2 ^{\frac{2}{3}} 3}{8} : - \frac{3}{2 ^{\frac{7}{3}}}

- \frac{2 ^{\frac{2}{3}} 3}{8}

因式分解数字:

8 = 2^{3}

= - \frac{2 ^{\frac{2}{3}} 3}{2 ^{3}}

化简

\frac{2 ^{\frac{2}{3}}}{2 ^{3}} : \frac{1}{2 ^{\frac{7}{3}}}

\frac{2 ^{\frac{2}{3}}}{2 ^{3}}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

= \frac{1}{2 ^{3 - \frac{2}{3}}}

3 - \frac{2}{3} = \frac{7}{3}

3 - \frac{2}{3}

将项转换为分式:

3 = \frac{3 \cdot 3}{3}

= \frac{3 \cdot 3}{3} - \frac{2}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 \cdot 3 - 2}{3}

3 \cdot 3 - 2 = 7

3 \cdot 3 - 2

数字相乘:

3 \cdot 3 = 9

= 9 - 2

数字相减:

9 - 2 = 7

= 7

= \frac{7}{3}

= \frac{1}{2 ^{\frac{7}{3}}}

= - \frac{3}{2 ^{\frac{7}{3}}}

2 u \cdot 0.54556 \dots = - \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 \cdot 0.54556 \dots

2 u \cdot 0.54556 \dots = - \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 \cdot 0.54556 \dots

\frac{2 u \cdot 0.54556 \dots}{2 \cdot 0.54556 \dots} = \frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 \cdot 0.54556 \dots}

化简

\frac{2 u \cdot 0.54556 \dots}{2 \cdot 0.54556 \dots} = \frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 \cdot 0.54556 \dots}

化简

\frac{2 u \cdot 0.54556 \dots}{2 \cdot 0.54556 \dots} : u

\frac{2 u \cdot 0.54556 \dots}{2 \cdot 0.54556 \dots}

约分:

2

= \frac{u \cdot 0.54556 \dots}{0.54556 \dots}

约分:

0.54556 \dots

= u

化简

\frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 \cdot 0.54556 \dots} : - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

\frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 \cdot 0.54556 \dots}

数字相乘:

2 \cdot 0.54556 \dots = 1.09112 \dots

= \frac{- \frac{3}{2 ^{\frac{7}{3}}}}{1.09112 \dots}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{3}{2 ^{\frac{7}{3}}}}{1.09112 \dots}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

\frac{\frac{3}{2 ^{\frac{7}{3}}}}{1.09112 \dots} = \frac{3}{2 ^{\frac{7}{3}} \cdot 1.09112 \dots}

= - \frac{3}{2 ^{\frac{7}{3}} \cdot 1.09112 \dots}

2^{\frac{7}{3}} = 2^{2} 32

2^{\frac{7}{3}}

2^{\frac{7}{3}} = 2^{2 + \frac{1}{3}}

= 2^{2 + \frac{1}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{2} \cdot 2^{\frac{1}{3}}

整理后得

= 2^{2} 32

= - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

对于

2 uv = - \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

，用

- 0.54556 \dots

替代

v : u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

对于

2 uv = - \frac{2 ^{\frac{2}{3}} \cdot 3}{8}

，用

- 0.54556 \dots

替代

v

2 u (- 0.54556 \dots) = - \frac{2 ^{\frac{2}{3}} 3}{8}

解

2 u (- 0.54556 \dots) = - \frac{2 ^{\frac{2}{3}} 3}{8} : u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

2 u (- 0.54556 \dots) = - \frac{2 ^{\frac{2}{3}} 3}{8}

化简

- \frac{2 ^{\frac{2}{3}} 3}{8} : - \frac{3}{2 ^{\frac{7}{3}}}

- \frac{2 ^{\frac{2}{3}} 3}{8}

因式分解数字:

8 = 2^{3}

= - \frac{2 ^{\frac{2}{3}} 3}{2 ^{3}}

化简

\frac{2 ^{\frac{2}{3}}}{2 ^{3}} : \frac{1}{2 ^{\frac{7}{3}}}

\frac{2 ^{\frac{2}{3}}}{2 ^{3}}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

= \frac{1}{2 ^{3 - \frac{2}{3}}}

3 - \frac{2}{3} = \frac{7}{3}

3 - \frac{2}{3}

将项转换为分式:

3 = \frac{3 \cdot 3}{3}

= \frac{3 \cdot 3}{3} - \frac{2}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3 \cdot 3 - 2}{3}

3 \cdot 3 - 2 = 7

3 \cdot 3 - 2

数字相乘:

3 \cdot 3 = 9

= 9 - 2

数字相减:

9 - 2 = 7

= 7

= \frac{7}{3}

= \frac{1}{2 ^{\frac{7}{3}}}

= - \frac{3}{2 ^{\frac{7}{3}}}

2 u (- 0.54556 \dots) = - \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 (- 0.54556 \dots)

2 u (- 0.54556 \dots) = - \frac{3}{2 ^{\frac{7}{3}}}

两边除以

2 (- 0.54556 \dots)

\frac{2 u ( - 0.54556 \dots )}{2 ( - 0.54556 \dots )} = \frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 ( - 0.54556 \dots )}

化简

\frac{2 u ( - 0.54556 \dots )}{2 ( - 0.54556 \dots )} = \frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 ( - 0.54556 \dots )}

化简

\frac{2 u ( - 0.54556 \dots )}{2 ( - 0.54556 \dots )} : u

\frac{2 u ( - 0.54556 \dots )}{2 ( - 0.54556 \dots )}

数字相乘:

2 (- 0.54556 \dots) = - 1.09112 \dots

= \frac{- 1.09112 \dots u}{- 1.09112 \dots}

约分:

- 1.09112 \dots

= u

化简

\frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 ( - 0.54556 \dots )} : \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

\frac{- \frac{3}{2 ^{\frac{7}{3}}}}{2 ( - 0.54556 \dots )}

去除括号:

(- a) = - a

= \frac{- \frac{3}{2 ^{\frac{7}{3}}}}{- 2 \cdot 0.54556 \dots}

数字相乘:

2 \cdot 0.54556 \dots = 1.09112 \dots

= \frac{- \frac{3}{2 ^{\frac{7}{3}}}}{- 1.09112 \dots}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{\frac{3}{2 ^{\frac{7}{3}}}}{1.09112 \dots}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3}{2 ^{\frac{7}{3}} \cdot 1.09112 \dots}

2^{\frac{7}{3}} = 2^{2} 32

2^{\frac{7}{3}}

2^{\frac{7}{3}} = 2^{2 + \frac{1}{3}}

= 2^{2 + \frac{1}{3}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 2^{2} \cdot 2^{\frac{1}{3}}

整理后得

= 2^{2} 32

= \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}

将解代入原方程进行验证

将它们代入

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

检验解是否符合

去除与方程不符的解。

检验

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = - 0.54556 \dots

的解:

真

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

代入

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = - 0.54556 \dots

(\frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots})^{2} - (- 0.54556 \dots)^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

整理后得

\frac{3 3 2}{38.09762 \dots} - 0.29763 \dots = - \frac{2 ^{\frac{2}{3}}}{8}

真

检验

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots

的解:

真

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

代入

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots

(- \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots})^{2} - 0.54556 \dots^{2} = - \frac{2 ^{\frac{2}{3}}}{8}

整理后得

\frac{3 3 2}{38.09762 \dots} - 0.29763 \dots = - \frac{2 ^{\frac{2}{3}}}{8}

真

将它们代入

2 uv = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

检验解是否符合

去除与方程不符的解。

检验

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = - 0.54556 \dots

的解:

真

2 uv = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

代入

u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = - 0.54556 \dots

2 \cdot \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} (- 0.54556 \dots) = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

整理后得

- \frac{3 \cdot 2 ^{\frac{2}{3}} \cdot 0.54556 \dots}{4.36449 \dots} = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

真

检验

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots

的解:

真

2 uv = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

代入

u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots

2 (- \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}) \cdot 0.54556 \dots = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

整理后得

- \frac{3 \cdot 2 ^{\frac{2}{3}} \cdot 0.54556 \dots}{4.36449 \dots} = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

真

因而，

u^{2} - v^{2} = - \frac{2 ^{\frac{2}{3}}}{8}, 2 uv = - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8}

最后的解是

(u = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, u = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots}, v = 0.54556 \dots v = - 0.54556 \dots)

w = u + v i

代回

w = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, w = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

解为

w = \frac{2 ^{\frac{2}{3}}}{2}, w = - \frac{2 ^{\frac{2}{3}}}{2}, w = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, w = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i, w = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, w = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

w = cos (x)

代回

cos (x) = \frac{2 ^{\frac{2}{3}}}{2}, cos (x) = - \frac{2 ^{\frac{2}{3}}}{2}, cos (x) = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, cos (x) = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i, cos (x) = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, cos (x) = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

cos (x) = \frac{2 ^{\frac{2}{3}}}{2}, cos (x) = - \frac{2 ^{\frac{2}{3}}}{2}, cos (x) = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, cos (x) = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i, cos (x) = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i, cos (x) = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

cos (x) = \frac{2 ^{\frac{2}{3}}}{2} : x = arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = 2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

cos (x) = \frac{2 ^{\frac{2}{3}}}{2}

使用反三角函数性质

cos (x) = \frac{2 ^{\frac{2}{3}}}{2}

cos (x) = \frac{2 ^{\frac{2}{3}}}{2}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = 2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

x = arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = 2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

cos (x) = - \frac{2 ^{\frac{2}{3}}}{2} : x = arccos (- \frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = - arccos (- \frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

cos (x) = - \frac{2 ^{\frac{2}{3}}}{2}

使用反三角函数性质

cos (x) = - \frac{2 ^{\frac{2}{3}}}{2}

cos (x) = - \frac{2 ^{\frac{2}{3}}}{2}

的通解

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = - arccos (- \frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

x = arccos (- \frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = - arccos (- \frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

cos (x) = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i :

无解

cos (x) = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i

化简

\frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i : \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots} + 0.54556 \dots i

\frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i

2^{2} 32 \cdot 1.09112 \dots = 32 \cdot 4.36449 \dots

2^{2} 32 \cdot 1.09112 \dots

2^{2} = 4

= 432 \cdot 1.09112 \dots

数字相乘:

4 \cdot 1.09112 \dots = 4.36449 \dots

= 32 \cdot 4.36449 \dots

= \frac{3}{3 2 \cdot 4.36449 \dots} + 0.54556 \dots i

\frac{3}{4.36449 \dots 3 2} = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots}

\frac{3}{4.36449 \dots 3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{4.36449 \dots 3 2 \cdot 2 ^{\frac{2}{3}}}

4.36449 \dots 32 \cdot 2^{\frac{2}{3}} = 8.72898 \dots

4.36449 \dots 32 \cdot 2^{\frac{2}{3}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 4.36449 \dots \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1}

使用法则

a^{1} = a

= 2

= 2 \cdot 4.36449 \dots

数字相乘:

4.36449 \dots \cdot 2 = 8.72898 \dots

= 8.72898 \dots

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots}

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots} + 0.54556 \dots i

无解

cos (x) = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i :

无解

cos (x) = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

化简

- \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i : - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots} - 0.54556 \dots i

- \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

2^{2} 32 \cdot 1.09112 \dots = 32 \cdot 4.36449 \dots

2^{2} 32 \cdot 1.09112 \dots

2^{2} = 4

= 432 \cdot 1.09112 \dots

数字相乘:

4 \cdot 1.09112 \dots = 4.36449 \dots

= 32 \cdot 4.36449 \dots

= - \frac{3}{3 2 \cdot 4.36449 \dots} - 0.54556 \dots i

\frac{3}{4.36449 \dots 3 2} = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots}

\frac{3}{4.36449 \dots 3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{4.36449 \dots 3 2 \cdot 2 ^{\frac{2}{3}}}

4.36449 \dots 32 \cdot 2^{\frac{2}{3}} = 8.72898 \dots

4.36449 \dots 32 \cdot 2^{\frac{2}{3}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 4.36449 \dots \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1}

使用法则

a^{1} = a

= 2

= 2 \cdot 4.36449 \dots

数字相乘:

4.36449 \dots \cdot 2 = 8.72898 \dots

= 8.72898 \dots

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots}

= - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots} - 0.54556 \dots i

无解

cos (x) = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i :

无解

cos (x) = - \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i

化简

- \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i : - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots} + 0.54556 \dots i

- \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} + 0.54556 \dots i

2^{2} 32 \cdot 1.09112 \dots = 32 \cdot 4.36449 \dots

2^{2} 32 \cdot 1.09112 \dots

2^{2} = 4

= 432 \cdot 1.09112 \dots

数字相乘:

4 \cdot 1.09112 \dots = 4.36449 \dots

= 32 \cdot 4.36449 \dots

= - \frac{3}{3 2 \cdot 4.36449 \dots} + 0.54556 \dots i

\frac{3}{4.36449 \dots 3 2} = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots}

\frac{3}{4.36449 \dots 3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{4.36449 \dots 3 2 \cdot 2 ^{\frac{2}{3}}}

4.36449 \dots 32 \cdot 2^{\frac{2}{3}} = 8.72898 \dots

4.36449 \dots 32 \cdot 2^{\frac{2}{3}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 4.36449 \dots \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1}

使用法则

a^{1} = a

= 2

= 2 \cdot 4.36449 \dots

数字相乘:

4.36449 \dots \cdot 2 = 8.72898 \dots

= 8.72898 \dots

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots}

= - \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots} + 0.54556 \dots i

无解

cos (x) = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i :

无解

cos (x) = \frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

化简

\frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i : \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots} - 0.54556 \dots i

\frac{3}{2 ^{2} 3 2 \cdot 1.09112 \dots} - 0.54556 \dots i

2^{2} 32 \cdot 1.09112 \dots = 32 \cdot 4.36449 \dots

2^{2} 32 \cdot 1.09112 \dots

2^{2} = 4

= 432 \cdot 1.09112 \dots

数字相乘:

4 \cdot 1.09112 \dots = 4.36449 \dots

= 32 \cdot 4.36449 \dots

= \frac{3}{3 2 \cdot 4.36449 \dots} - 0.54556 \dots i

\frac{3}{4.36449 \dots 3 2} = \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots}

\frac{3}{4.36449 \dots 3 2}

乘以共轭根式

\frac{2 ^{\frac{2}{3}}}{2 ^{\frac{2}{3}}}

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{4.36449 \dots 3 2 \cdot 2 ^{\frac{2}{3}}}

4.36449 \dots 32 \cdot 2^{\frac{2}{3}} = 8.72898 \dots

4.36449 \dots 32 \cdot 2^{\frac{2}{3}}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

2^{\frac{2}{3}} 32 = 2^{\frac{2}{3}} \cdot 2^{\frac{1}{3}} = 2^{\frac{2}{3} + \frac{1}{3}}

= 4.36449 \dots \cdot 2^{\frac{2}{3} + \frac{1}{3}}

2^{\frac{2}{3} + \frac{1}{3}} = 2

2^{\frac{2}{3} + \frac{1}{3}}

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1}

使用法则

a^{1} = a

= 2

= 2 \cdot 4.36449 \dots

数字相乘:

4.36449 \dots \cdot 2 = 8.72898 \dots

= 8.72898 \dots

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots}

= \frac{3 \cdot 2 ^{\frac{2}{3}}}{8.72898 \dots} - 0.54556 \dots i

无解

合并所有解

x = arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = 2 π - arccos (\frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = arccos (- \frac{2 ^{\frac{2}{3}}}{2}) + 2 πn, x = - arccos (- \frac{2 ^{\frac{2}{3}}}{2}) + 2 πn

以小数形式表示解

x = 0.88929 \dots + 2 πn, x = 2 π - 0.88929 \dots + 2 πn, x = 2.25229 \dots + 2 πn, x = - 2.25229 \dots + 2 πn

作图

流行的例子

solvefor w,s(t)=Ae^{-ct}cos(wt+θ)

so l v e f or w, s (t) = A e^{- c t} cos (wt + θ)

sqrt(2)=2sin(x)

2 = 2 sin (x)

6 sin (\frac{π}{4} x) = 3

sec(x)=3,(3pi)/2 <= x<= 2pi,sin(2x)

sec (x) = 3, \frac{3 π}{2} \leq x \leq 2 π, sin (2 x)

2+8cos(θ)=-1+2cos(θ)

2 + 8 cos (θ) = - 1 + 2 cos (θ)