[출제의도] 유리수와 무리수가 들어있는 식을 간단히 하여 식의 값을 계산할 수 있는가를 묻는 문제이다.
x = 1 1 − 1 1 + 1 3 7 − 1 = 1 1 − 1 ( 3 7 − 1 ) + 1 3 7 − 1 \displaystyle x = \frac{1}{1 - \frac{1}{1 + \frac{1}{3 \sqrt{7} - 1}}} = \frac{1}{1 - \frac{1}{\frac{( 3 \sqrt{7} - 1 ) + 1}{3 \sqrt{7} - 1}}} x = 1 − 1 + 3 7 − 1 1 1 1 = 1 − 3 7 − 1 ( 3 7 − 1 ) + 1 1 1 = 1 1 − 3 7 − 1 3 7 = 1 3 7 − ( 3 7 − 1 ) 3 7 = 3 7 \displaystyle = \frac{1}{1 - \frac{3 \sqrt{7} - 1}{3 \sqrt{7}}} = \frac{1}{\frac{3 \sqrt{7} - ( 3 \sqrt{7} - 1 )}{3 \sqrt{7}}} = 3 \sqrt{7} = 1 − 3 7 3 7 − 1 1 = 3 7 3 7 − ( 3 7 − 1 ) 1 = 3 7 x 2 = ( 3 7 ) 2 = 63 \displaystyle x ^{2} = ( 3 \sqrt{7} ) ^{2} = 63 x 2 = ( 3 7 ) 2 = 63 3 7 − 1 = t \displaystyle 3 \sqrt{7} - 1 = t 3 7 − 1 = t x = 1 1 − 1 1 + 1 t \displaystyle x = \frac{1}{1 - \frac{1}{1 + \frac{1}{t}}} x = 1 − 1 + t 1 1 1 = 1 1 − 1 t + 1 t \displaystyle = \frac{1}{1 - \frac{1}{\frac{t + 1}{t}}} = 1 − t t + 1 1 1 = 1 1 − t t + 1 \displaystyle = \frac{1}{1 - \frac{t}{t + 1}} = 1 − t + 1 t 1 = 1 t + 1 − t t + 1 = t + 1 \displaystyle = \frac{1}{\frac{t + 1 - t}{t + 1}} = t + 1 = t + 1 t + 1 − t 1 = t + 1 x = t + 1 = 3 7 \displaystyle x = t + 1 = 3 \sqrt{7} x = t + 1 = 3 7 x 2 = ( 3 7 ) 2 = 63 \displaystyle x ^{2} = ( 3 \sqrt{7} ) ^{2} = 63 x 2 = ( 3 7 ) 2 = 63