与えられた6つの計算問題を解く問題です。 (1) $8^{\frac{2}{3}} \times 4^{\frac{3}{2}}$ (2) $2^{-\frac{1}{2}} \times 2^{\frac{5}{6}} \div 2^{\frac{1}{3}}$ (3) $(3^{-2} \times 9^{\frac{2}{3}})^{\frac{3}{2}}$ (4) $\sqrt[4]{4} \times \sqrt[8]{8}$ (5) $\sqrt[3]{5} \div \sqrt[12]{5} \times \sqrt[8]{25}$ (6) $\sqrt{6} \times \sqrt[4]{54} \div \sqrt{6}$

1. 問題の内容

与えられた6つの計算問題を解く問題です。

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

(2) $2^{-\frac{1}{2}} \times 2^{\frac{5}{6}} \div 2^{\frac{1}{3}}$

(3) $(3^{-2} \times 9^{\frac{2}{3}})^{\frac{3}{2}}$

(4) $\sqrt[4]{4} \times \sqrt[8]{8}$

(5) $\sqrt[3]{5} \div \sqrt[12]{5} \times \sqrt[8]{25}$

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

2. 解き方の手順

(1)

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

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

$4 \times 8 = 32$

(2)

$2^{-\frac{1}{2}} \times 2^{\frac{5}{6}} \div 2^{\frac{1}{3}} = 2^{-\frac{1}{2} + \frac{5}{6} - \frac{1}{3}} = 2^{-\frac{3}{6} + \frac{5}{6} - \frac{2}{6}} = 2^{\frac{0}{6}} = 2^0 = 1$

(3)

$(3^{-2} \times 9^{\frac{2}{3}})^{\frac{3}{2}} = (3^{-2} \times (3^2)^{\frac{2}{3}})^{\frac{3}{2}} = (3^{-2} \times 3^{\frac{4}{3}})^{\frac{3}{2}} = (3^{-2 + \frac{4}{3}})^{\frac{3}{2}} = (3^{-\frac{6}{3} + \frac{4}{3}})^{\frac{3}{2}} = (3^{-\frac{2}{3}})^{\frac{3}{2}} = 3^{-1} = \frac{1}{3}$

(4)

$\sqrt[4]{4} \times \sqrt[8]{8} = 4^{\frac{1}{4}} \times 8^{\frac{1}{8}} = (2^2)^{\frac{1}{4}} \times (2^3)^{\frac{1}{8}} = 2^{\frac{2}{4}} \times 2^{\frac{3}{8}} = 2^{\frac{1}{2}} \times 2^{\frac{3}{8}} = 2^{\frac{1}{2} + \frac{3}{8}} = 2^{\frac{4}{8} + \frac{3}{8}} = 2^{\frac{7}{8}}$

(5)

$\sqrt[3]{5} \div \sqrt[12]{5} \times \sqrt[8]{25} = 5^{\frac{1}{3}} \div 5^{\frac{1}{12}} \times (5^2)^{\frac{1}{8}} = 5^{\frac{1}{3} - \frac{1}{12} + \frac{2}{8}} = 5^{\frac{1}{3} - \frac{1}{12} + \frac{1}{4}} = 5^{\frac{4}{12} - \frac{1}{12} + \frac{3}{12}} = 5^{\frac{6}{12}} = 5^{\frac{1}{2}} = \sqrt{5}$

(6)

$\sqrt{6} \times \sqrt[4]{54} \div \sqrt{6} = \sqrt[4]{54} = \sqrt[4]{2 \times 3^3} = (2 \times 3^3)^{\frac{1}{4}} = (2 \times 27)^{\frac{1}{4}} = \sqrt[4]{54}$

$\sqrt{6} \times \sqrt[4]{54} \div \sqrt{6} = \sqrt[4]{54}$

$\sqrt[4]{54} = \sqrt[4]{2\cdot3^3} = \sqrt[4]{2\cdot3^3}$

3. 最終的な答え

(1) 32

(2) 1

(3) $\frac{1}{3}$

(4) $2^{\frac{7}{8}}$

(5) $\sqrt{5}$

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