与えられた6つの式をそれぞれ簡単にせよという問題です。式は以下の通りです。 (1) $\sqrt[4]{27\sqrt[4]{3}}$ (2) $\frac{\sqrt[3]{24}}{\sqrt[3]{3}}$ (3) $(\sqrt[3]{6})^2\sqrt[3]{48}$ (4) $\frac{(\sqrt[4]{54})^3}{\sqrt[4]{24}}$ (5) $\frac{\sqrt[3]{147}\sqrt[3]{63}}{7}$ (6) $\sqrt[3]{\sqrt{0.005}\sqrt[3]{0.025}}$

1. 問題の内容

与えられた6つの式をそれぞれ簡単にせよという問題です。式は以下の通りです。

(1) $\sqrt[4]{27\sqrt[4]{3}}$

(2) $\frac{\sqrt[3]{24}}{\sqrt[3]{3}}$

(3) $(\sqrt[3]{6})^2\sqrt[3]{48}$

(4) $\frac{(\sqrt[4]{54})^3}{\sqrt[4]{24}}$

(5) $\frac{\sqrt[3]{147}\sqrt[3]{63}}{7}$

(6) $\sqrt[3]{\sqrt{0.005}\sqrt[3]{0.025}}$

2. 解き方の手順

(1) $\sqrt[4]{27\sqrt[4]{3}} = \sqrt[4]{3^3 \cdot 3^{\frac{1}{4}}} = \sqrt[4]{3^{\frac{13}{4}}} = (3^{\frac{13}{4}})^{\frac{1}{4}} = 3^{\frac{13}{16}} = \sqrt[16]{3^{13}}$

(2) $\frac{\sqrt[3]{24}}{\sqrt[3]{3}} = \sqrt[3]{\frac{24}{3}} = \sqrt[3]{8} = 2$

(3) $(\sqrt[3]{6})^2\sqrt[3]{48} = \sqrt[3]{6^2} \cdot \sqrt[3]{48} = \sqrt[3]{36 \cdot 48} = \sqrt[3]{36 \cdot 12 \cdot 4} = \sqrt[3]{6^2 \cdot 6 \cdot 2 \cdot 2^2} = \sqrt[3]{6^3 \cdot 2^3} = 6 \cdot 2 = 12$

(4) $\frac{(\sqrt[4]{54})^3}{\sqrt[4]{24}} = \sqrt[4]{\frac{54^3}{24}} = \sqrt[4]{\frac{(2 \cdot 3^3)^3}{2^3 \cdot 3}} = \sqrt[4]{\frac{2^3 \cdot 3^9}{2^3 \cdot 3}} = \sqrt[4]{3^8} = 3^2 = 9$

(5) $\frac{\sqrt[3]{147}\sqrt[3]{63}}{7} = \frac{\sqrt[3]{3 \cdot 49}\sqrt[3]{9 \cdot 7}}{7} = \frac{\sqrt[3]{3 \cdot 7^2} \sqrt[3]{3^2 \cdot 7}}{7} = \frac{\sqrt[3]{3^3 \cdot 7^3}}{7} = \frac{3 \cdot 7}{7} = 3$

(6) $\sqrt[3]{\sqrt{0.005}\sqrt[3]{0.025}} = \sqrt[3]{\sqrt{\frac{5}{1000}} \sqrt[3]{\frac{25}{1000}}} = \sqrt[3]{\sqrt{\frac{1}{200}} \sqrt[3]{\frac{1}{40}}} = \sqrt[3]{(\frac{1}{200})^{\frac{1}{2}} (\frac{1}{40})^{\frac{1}{3}}} = \sqrt[3]{\frac{1}{\sqrt{200}} \cdot \frac{1}{\sqrt[3]{40}}} = \sqrt[3]{\frac{1}{10\sqrt{2}}\frac{1}{\sqrt[3]{40}}} = \sqrt[3]{\frac{1}{10\sqrt{2} \sqrt[3]{40}}} = \sqrt[3]{\frac{1}{10\sqrt{2} \sqrt[3]{8 \cdot 5}}} = \sqrt[3]{\frac{1}{10\sqrt{2} 2\sqrt[3]{5}}} = \sqrt[3]{\frac{1}{20 \sqrt{2} \sqrt[3]{5}}} = \sqrt[3]{\frac{1}{20 \cdot 2^{1/2} 5^{1/3}}}= (\frac{1}{20 \cdot 2^{1/2} 5^{1/3}})^{\frac{1}{3}} = (\frac{1}{2^2 \cdot 5 \cdot 2^{1/2} 5^{1/3}})^{\frac{1}{3}} = (\frac{1}{2^{5/2} 5^{4/3}})^{\frac{1}{3}} = \frac{1}{2^{\frac{5}{6}} 5^{\frac{4}{9}}}$

$0.005 = \frac{5}{1000} = \frac{1}{200}$

$0.025 = \frac{25}{1000} = \frac{1}{40}$

\sqrt[3]{\sqrt{\frac{1}{200}} \sqrt[3]{\frac{1}{40}}} = \sqrt[3]{(\frac{1}{200})^{1/2} (\frac{1}{40})^{1/3}} = \sqrt[3]{(\frac{1}{2^3 \cdot 5^2})^{1/2} (\frac{1}{2^3 \cdot 5})^{1/3}} = \sqrt[3]{(\frac{1}{2^{3/2} 5})^{1/2} (\frac{1}{2 \cdot 5^{1/3}})} = \frac{1}{\sqrt[3]{\sqrt{\frac{1}{200}}\sqrt[3]{\frac{1}{40}}}} = \frac{1}{\sqrt[3]{ \frac{1}{10\sqrt{2}} \cdot \frac{1}{\sqrt[3]{40}}}} = \frac{1}{\sqrt[3]{\frac{1}{10\sqrt{2} \sqrt[3]{8 \cdot 5}}}} = \frac{1}{\sqrt[3]{\frac{1}{20\sqrt{2}\sqrt[3]{5}}}} = \frac{1}{\sqrt[3]{\frac{\sqrt{2}}{40\sqrt[3]{5}\sqrt{2}}}} = \frac{1}{\sqrt[3]{\frac{\sqrt{2}}{40\sqrt[3]{10}}}} = \sqrt[3]{\frac{1}{8 \cdot 250}} = \frac{1}{\sqrt[3]{40\sqrt[2]{2}\sqrt[3]{5}}} = \frac{1}{\sqrt[3]{\frac{1}{200}}^{\frac{1}{2}} \sqrt[3]{\frac{1}{40}}}} = \sqrt[18]{\frac{2^3*25}{4*625}}=\frac{\sqrt[3]{\sqrt[1]{2}}}{\sqrt[3]{\frac{25}{4}}}

$\sqrt[3]{\sqrt{0.005}\sqrt[3]{0.025}} = \sqrt[3]{\sqrt{\frac{5}{1000}} \sqrt[3]{\frac{25}{1000}}} = \sqrt[3]{(\frac{5}{1000})^{1/2} (\frac{25}{1000})^{1/3}} = \sqrt[3]{(\frac{1}{200})^{1/2} (\frac{1}{40})^{1/3}} = \sqrt[3]{\frac{1}{\sqrt{200}} \cdot \frac{1}{\sqrt[3]{40}}} = \sqrt[3]{\frac{1}{10\sqrt{2} \cdot 2\sqrt[3]{5}}} = \sqrt[3]{\frac{1}{20\sqrt{2}\sqrt[3]{5}}} = \frac{1}{20^{\frac{1}{3}} 2^{\frac{1}{6}} 5^{\frac{1}{9}}}$

3. 最終的な答え

(1) $\sqrt[16]{3^{13}}$

(2) $2$

(3) $12$

(4) $9$

(5) $3$

(6) $\sqrt[18]{\frac{1}{200}\cdot \frac{1}{40}} = \sqrt[3]{\sqrt{0.005}\sqrt[3]{0.025}}$