簡約 32^{1/3}

解

簡約

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

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

十進法表記

3.17480 \dots

解答ステップ

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

以下の素因数分解:

32 : 2^{5}

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

指数の規則を適用する:

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

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

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

指数の規則を適用する:

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

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

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

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

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

指数の規則を適用する:

(a^{b})^{c} = a^{b c}, a \geq 0

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

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

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

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