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指数の計算の問題の解法 | 夢を叶える塾

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指数の計算の問題の解法

2017/10/22 2019/1/18 指数関数

指数の計算の法則を使った計算問題です。

見出し

1 基本問題
- 1.1 解き方
  - 1.1.1 解説
2 終わりに

基本問題

$\sqrt[3]{6}\sqrt[3]{12}$

解き方

指数の計算の法則を使います。

$a^ma^n=a^{m+n}$
$(a^m)^n=a^{mn}$
$(ab)^n=a^nb^n$

また、n乗根は $\displaystyle \frac{1}{n}$ 乗になります。
$\displaystyle \sqrt[n]{a}=a^{\frac{1}{n}}$

解説

$\sqrt[3]{6}\sqrt[3]{12}$
まず3乗根の中を素因数分解して計算しやすくします。
$=\sqrt[3]{2\cdot 3}\sqrt[3]{2^2\cdot 3}$
次に指数の計算法則の $(ab)^n=a^nb^n$ を右から左に使います。
ちなみに $\sqrt[n]{a}=a^{1/n}$ なのでこの法則が使えるわけですね。
$=\sqrt[3]{(2\cdot 3)\cdot (2^2\cdot 3)}$
$=\sqrt[3]{2^3\cdot 3^2}$
$\displaystyle \sqrt{2^2}=(2^2)^{\frac{1}{2}}=2^{\frac{2}{2}}=2^1=2$ としたように $\displaystyle \sqrt[3]{2^3}=(2^3)^{\frac{1}{3}}=2^{\frac{3}{3}}=2^1=2$ とできます。
$=\sqrt[3]{2^3}\cdot \sqrt[3]{3^2}$
$=2\sqrt[3]{3^2}$

終わりに

指数の計算の法則を覚えるとともに、 $\displaystyle \sqrt[n]{a}=a^{\frac{1}{n}}$ であることも覚えておきましょう。

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