簡約 (sqrt(16^3))^{1/4}

解

簡約

(1 6^{3})^{\frac{1}{4}}

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

十進法表記

2.82842 \dots

解答ステップ

(1 6^{3})^{\frac{1}{4}}

1 6^{3} = 64

= 6 4^{\frac{1}{4}}

以下の素因数分解:

64 : 2^{6}

= (2^{6})^{\frac{1}{4}}

指数の規則を適用する:

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

2^{6} = 2^{4} \cdot 2^{2}

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

指数の規則を適用する:

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

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

= (2^{4})^{\frac{1}{4}} (2^{2})^{\frac{1}{4}}

(2^{4})^{\frac{1}{4}} = 2

= 2 (2^{2})^{\frac{1}{4}}

指数の規則を適用する:

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

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

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

2 \cdot \frac{1}{4} = \frac{1}{2}

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