簡約 \sqrt[3]{\sqrt[4]{3^5*3}}

解

簡約

3 4 3^{5} \cdot 3

3

十進法表記

1.73205 \dots

解答ステップ

3 4 3^{5} \cdot 3

4 3^{5} \cdot 3 = 33

= 333

累乗根の規則を適用する:

n ab = n a n b, a \geq 0, b \geq 0

333 = 3333

= 3333

33 = 63

= 3363

累乗根の規則を適用する:

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

33 = 3^{\frac{1}{3}}

= 3^{\frac{1}{3}} 63

累乗根の規則を適用する:

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

63 = 3^{\frac{1}{6}}

= 3^{\frac{1}{3}} \cdot 3^{\frac{1}{6}}

指数の規則を適用する:

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

3^{\frac{1}{3}} \cdot 3^{\frac{1}{6}} = 3^{\frac{1}{3} + \frac{1}{6}}

= 3^{\frac{1}{3} + \frac{1}{6}}

\frac{1}{3} + \frac{1}{6} = \frac{1}{2}

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

指数の規則を適用する:

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

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

= 3