거듭제곱근 대소

[출제의도] 거듭제곱근의 대소를 비교하여 식을 간단히 할 수 있는가를 묻는 문제이다. $\displaystyle \sqrt{2} = 2 ^{\frac{1}{2}}$, $\displaystyle \sqrt[3]{3} = 3 ^{\frac{1}{3}}$, $\displaystyle \sqrt[5]{5} = 5 ^{\frac{1}{5}}$ $\displaystyle 2 ^{\frac{1}{2}}$, $\displaystyle 3 ^{\frac{1}{3}}$에서 $\displaystyle \left( 2 ^{\frac{1}{2}} \right) ^{6} = 2 ^{3} = 8$, $\displaystyle \left( 3 ^{\frac{1}{3}} \right) ^{6} = 3 ^{2} = 9$ ∴ $\displaystyle \sqrt{2} < \sqrt[3]{3}$ $\cdots \text{㉠}$ $\displaystyle 2 ^{\frac{1}{2}}$, $\displaystyle 5 ^{\frac{1}{5}}$에서 $\displaystyle \left( 2 ^{\frac{1}{2}} \right) ^{10} = 2 ^{5} = 32$, $\displaystyle \left( 5 ^{\frac{1}{5}} \right) ^{10} = 5 ^{2} = 25$ ∴ $\displaystyle \sqrt[5]{5} < \sqrt{2}$$\cdots \text{㉡}$ ㉠, ㉡에서 $\displaystyle \sqrt[5]{5} < \sqrt{2} < \sqrt[3]{3}$ ∴ $\displaystyle | \sqrt{2} - \sqrt[3]{3} | + | \sqrt[3]{3} - \sqrt[5]{5} | + | \sqrt[5]{5} - \sqrt{2} |$ $\displaystyle = - ( \sqrt{2} - \sqrt[3]{3} ) + ( \sqrt[3]{3} - \sqrt[5]{5} ) - ( \sqrt[5]{5} - \sqrt{2} )$ $\displaystyle = 2 ( \sqrt[3]{3} - \sqrt[5]{5} )$