簡約 \sqrt[5]{(2^4*3)}^2

解

簡約

5 (2^{4} \cdot 3)^{2}

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

十進法表記

4.70431 \dots

解答ステップ

(5 (2^{4} \cdot 3))^{2}

累乗根の規則を適用する:

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

= (((2^{4} \cdot 3))^{\frac{1}{5}})^{2}

指数の規則を適用する:

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

= ((2^{4} \cdot 3))^{\frac{1}{5} \cdot 2}

\frac{1}{5} \cdot 2 = \frac{2}{5}

= ((2^{4} \cdot 3))^{\frac{2}{5}}

指数の規則を適用する:

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

= 3^{\frac{2}{5}} (2^{4})^{\frac{2}{5}}

(2^{4})^{\frac{2}{5}} : 2^{\frac{8}{5}}

= 3^{\frac{2}{5}} \cdot 2^{\frac{8}{5}}

2^{\frac{8}{5}} = 2 \cdot 2^{\frac{3}{5}}

= 2 \cdot 3^{\frac{2}{5}} \cdot 2^{\frac{3}{5}}