지수 로그 식의 값

$\displaystyle 9 ^{a} = 2 ^{\frac{1}{b}}$에서 $3 ^{2a} = 2$, $( a + b ) ^{2} = \log _{3} 64$에서 $3 ^{( a + b ) ^{2}} = 64 = 2 ^{6}$, 즉 $\displaystyle 3 ^{\frac{( a + b ) ^{2}}{6}} = 2$ $\displaystyle 2 a = \frac{( a + b ) ^{2}}{6}$, $a ^{2} - 10 ab + b ^{2} = 0$, $\displaystyle t = \frac{a}{b}$라 하면 $a > b$$> 0$에서 $t > 1$이고 $\displaystyle \left( \frac{a}{b} \right) ^{2} - 10 \left( \frac{a}{b} \right) + 1 = 0$, $t ^{2} - 10 t + 1 = 0$, $\displaystyle t = 5 + 2 \sqrt{6}$ $\displaystyle \frac{a - b}{a + b} = \frac{t - 1}{t + 1}$$\displaystyle = \frac{4 + 2 \sqrt{6}}{6 + 2 \sqrt{6}}$$\displaystyle = \frac{2 + \sqrt{6}}{3 + \sqrt{6}}$ $\displaystyle = \frac{( 2 + \sqrt{6} ) ( 3 - \sqrt{6} )}{( 3 + \sqrt{6} ) ( 3 - \sqrt{6} )}$$\displaystyle = \frac{\sqrt{6}}{3}$