以下の3つの計算問題を解く。 (1) $2^0 \div 2^{-4}$ (3) $0.125^{-\frac{2}{3}}$ (5) $\sqrt{6} \times \sqrt[4]{54} \div \sqrt[4]{6}$

1. 問題の内容

以下の3つの計算問題を解く。

(1) $2^0 \div 2^{-4}$

(3) $0.125^{-\frac{2}{3}}$

(5) $\sqrt{6} \times \sqrt[4]{54} \div \sqrt[4]{6}$

2. 解き方の手順

(1) $2^0 \div 2^{-4}$ の計算

$2^0 = 1$

$2^{-4} = \frac{1}{2^4} = \frac{1}{16}$

$2^0 \div 2^{-4} = 1 \div \frac{1}{16} = 1 \times 16 = 16$

(3) $0.125^{-\frac{2}{3}}$ の計算

$0.125 = \frac{125}{1000} = \frac{1}{8}$

$0.125^{-\frac{2}{3}} = (\frac{1}{8})^{-\frac{2}{3}} = (8^{-1})^{-\frac{2}{3}} = 8^{\frac{2}{3}}$

$8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = 2^2 = 4$

(5) $\sqrt{6} \times \sqrt[4]{54} \div \sqrt[4]{6}$ の計算

$\sqrt{6} = 6^{\frac{1}{2}} = 6^{\frac{2}{4}} = \sqrt[4]{6^2} = \sqrt[4]{36}$

$\sqrt[4]{54} = \sqrt[4]{2 \times 27} = \sqrt[4]{2 \times 3^3}$

$\sqrt{6} \times \sqrt[4]{54} \div \sqrt[4]{6} = \sqrt[4]{36} \times \sqrt[4]{54} \div \sqrt[4]{6} = \frac{\sqrt[4]{36 \times 54}}{\sqrt[4]{6}} = \sqrt[4]{\frac{36 \times 54}{6}} = \sqrt[4]{\frac{6 \times 6 \times 6 \times 9}{6}} = \sqrt[4]{6 \times 6 \times 9} = \sqrt[4]{36 \times 9} = \sqrt[4]{324}$

$324 = 4 \times 81 = 2^2 \times 3^4$

$\sqrt[4]{324} = \sqrt[4]{2^2 \times 3^4} = 3 \times \sqrt[4]{4} = 3 \times \sqrt{2}$

3. 最終的な答え

(1) 16

(3) 4

(5) $3 \sqrt{2}$