簡約 1000^{3/2}

解

簡約

100 0^{\frac{3}{2}}

10000 \cdot 1 0^{\frac{1}{2}}

十進法表記

31622.77660 \dots

解答ステップ

100 0^{\frac{3}{2}}

以下の素因数分解:

1000 : 2^{3} \cdot 5^{3}

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

指数の規則を適用する:

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

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

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

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

指数の規則を適用する:

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

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

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

(2^{2})^{\frac{3}{2}} = 8

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

(5^{2})^{\frac{3}{2}} = 125

= 8 \cdot 125 (2 \cdot 5)^{\frac{3}{2}}

(2 \cdot 5)^{\frac{3}{2}} = 10 \cdot 1 0^{\frac{1}{2}}

= 8 \cdot 125 \cdot 10 \cdot 1 0^{\frac{1}{2}}

数を乗じる:

8 \cdot 125 \cdot 10 = 10000

= 10000 \cdot 1 0^{\frac{1}{2}}