与えられた各式を簡単にし、計算せよ。問題は全部で25問あります。

はい、承知いたしました。画像にある数学の問題を解いていきます。

1. 問題の内容

与えられた各式を簡単にし、計算せよ。問題は全部で25問あります。

2. 解き方の手順

(1) $6^{-1} = \frac{1}{6}$

(2) $8^{-2} = \frac{1}{8^2} = \frac{1}{64}$

(3) $3^{-3} = \frac{1}{3^3} = \frac{1}{27}$

(4) $10^{-4} = \frac{1}{10^4} = \frac{1}{10000}$

(5) $0.2^0 = 1$ (ゼロでない数 $x$ に対して、$x^0 = 1$)

(6) $(-5)^{-2} = \frac{1}{(-5)^2} = \frac{1}{25}$

(7) $\sqrt{36} = 6$

(8) $\sqrt{2}\sqrt{8} = \sqrt{2 \cdot 8} = \sqrt{16} = 4$

(9) $\frac{\sqrt{48}}{\sqrt{12}} = \sqrt{\frac{48}{12}} = \sqrt{4} = 2$

(10) $(\sqrt{3})^4 = (\sqrt{3}^2)^2 = (3)^2 = 9$

(11) $\frac{\sqrt{108}}{\sqrt{3}} = \sqrt{\frac{108}{3}} = \sqrt{36} = 6$

(12) $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

(13) $4^{\frac{1}{2}} = \sqrt{4} = 2$

(14) $27^{\frac{1}{3}} = \sqrt[3]{27} = 3$

(15) $100^{\frac{3}{2}} = (100^{\frac{1}{2}})^3 = (\sqrt{100})^3 = (10)^3 = 1000$

(16) $16^{0.75} = 16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 = (\sqrt[4]{16})^3 = (2)^3 = 8$

(17) $25^{-\frac{1}{2}} = \frac{1}{25^{\frac{1}{2}}} = \frac{1}{\sqrt{25}} = \frac{1}{5}$

(18) $(a^6)^5 = a^{6 \cdot 5} = a^{30}$

(19) $(a^3b^2)^5 = (a^3)^5 (b^2)^5 = a^{3 \cdot 5}b^{2 \cdot 5} = a^{15}b^{10}$

(20) $(-5)^{-2} = \frac{1}{(-5)^2} = \frac{1}{25}$

(21) $4^{-6} \times 4^5 \div 4^{-3} = 4^{-6+5-(-3)} = 4^{-6+5+3} = 4^{2} = 16$

(22) $\sqrt[3]{216} = 6$ (なぜなら、$6 \times 6 \times 6 = 216$ )

(23) $\sqrt[5]{160} \div \sqrt[5]{5} = \sqrt[5]{\frac{160}{5}} = \sqrt[5]{32} = 2$

(24) $\sqrt[3]{2} \times \sqrt[6]{16} = \sqrt[3]{2} \times \sqrt[6]{2^4} = 2^{1/3} \times 2^{4/6} = 2^{1/3} \times 2^{2/3} = 2^{1/3 + 2/3} = 2^1 = 2$

(25) $\sqrt[3]{3^8} \div \sqrt[3]{9} = \sqrt[3]{\frac{3^8}{9}} = \sqrt[3]{\frac{3^8}{3^2}} = \sqrt[3]{3^6} = 3^{\frac{6}{3}} = 3^2 = 9$

3. 最終的な答え

(1) $\frac{1}{6}$

(2) $\frac{1}{64}$

(3) $\frac{1}{27}$

(4) $\frac{1}{10000}$

(5) 1

(6) $\frac{1}{25}$

(7) 6

(8) 4

(9) 2

(10) 9

(11) 6

(12) $\frac{1}{9}$

(13) 2

(14) 3

(15) 1000

(16) 8

(17) $\frac{1}{5}$

(18) $a^{30}$

(19) $a^{15}b^{10}$

(20) $\frac{1}{25}$

(21) 16

(22) 6

(23) 2

(24) 2

(25) 9