簡約 (3.775)^{3/2}

解

簡約

(3.775)^{\frac{3}{2}}

\frac{151 \cdot 15 1 ^{\frac{1}{2}}}{80 \cdot 1 0 ^{\frac{1}{2}}}

十進法表記

7.33458 \dots

解答ステップ

3.77 5^{\frac{3}{2}}

3.775 = \frac{151}{40}

= (\frac{151}{40})^{\frac{3}{2}}

指数の規則を適用する:

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

= \frac{15 1 ^{\frac{3}{2}}}{4 0 ^{\frac{3}{2}}}

4 0^{\frac{3}{2}} = (2 \cdot 1 0^{\frac{1}{2}})^{3}

= \frac{15 1 ^{\frac{3}{2}}}{( 2 \cdot 1 0 ^{\frac{1}{2}} ) ^{3}}

15 1^{\frac{3}{2}} = 151 \cdot 15 1^{\frac{1}{2}}

= 151 \cdot 15 1^{\frac{1}{2}}

指数の規則を適用する:

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

(2 \cdot 1 0^{\frac{1}{2}})^{3} = 2^{3} (1 0^{\frac{1}{2}})^{3}

= \frac{151 \cdot 15 1 ^{\frac{1}{2}}}{2 ^{3} ( 1 0 ^{\frac{1}{2}} ) ^{3}}

(1 0^{\frac{1}{2}})^{3} : 1 0^{\frac{3}{2}}

= \frac{151 \cdot 15 1 ^{\frac{1}{2}}}{2 ^{3} \cdot 1 0 ^{\frac{3}{2}}}

1 0^{\frac{3}{2}} = 10 \cdot 1 0^{\frac{1}{2}}

= \frac{151 \cdot 15 1 ^{\frac{1}{2}}}{2 ^{3} \cdot 10 \cdot 1 0 ^{\frac{1}{2}}}

簡素化

2^{3} \cdot 10 \cdot 1 0^{\frac{1}{2}} : 5 \cdot 5^{\frac{1}{2}} \cdot 2^{\frac{1}{2} + 3 + 1}

= \frac{151 \cdot 15 1 ^{\frac{1}{2}}}{5 \cdot 5 ^{\frac{1}{2}} \cdot 2 ^{\frac{1}{2} + 3 + 1}}

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

= \frac{151 \cdot 15 1 ^{\frac{1}{2}}}{2 ^{4} \cdot 5 \cdot 2 ^{\frac{1}{2}} \cdot 5 ^{\frac{1}{2}}}

簡素化

5 \cdot 2^{4} \cdot 2^{\frac{1}{2}} \cdot 5^{\frac{1}{2}} : 2^{4} \cdot 5 \cdot 1 0^{\frac{1}{2}}

= \frac{151 \cdot 15 1 ^{\frac{1}{2}}}{2 ^{4} \cdot 5 \cdot 1 0 ^{\frac{1}{2}}}

5 \cdot 2^{4} \cdot 1 0^{\frac{1}{2}} = 80 \cdot 1 0^{\frac{1}{2}}

= \frac{151 \cdot 15 1 ^{\frac{1}{2}}}{80 \cdot 1 0 ^{\frac{1}{2}}}