簡約 243^{1/3}

解

簡約

24 3^{\frac{1}{3}}

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

十進法表記

6.24025 \dots

解答ステップ

24 3^{\frac{1}{3}}

以下の素因数分解:

243 : 3^{5}

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

指数の規則を適用する:

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

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

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

指数の規則を適用する:

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

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

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

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

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

指数の規則を適用する:

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

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

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

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

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