Question : If $x^{\frac{1}{4}}+x^{\frac{-1}{4}}=2$, then what is the value of $x^{81}+\frac{1}{x^{81}}$?
Option 1: –2
Option 2: 0
Option 3: 1
Option 4: 2
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Correct Answer: 2
Solution : Given: $x^{\frac{1}{4}}$ + $x^{-\frac{1}{4}}=2$ $x^{\frac{1}{4}}$ + $\frac{1}{x^{\frac{1}{4}}}=2$ ⇒ $x^{\frac{1}{4}}$ + $\frac{1}{x^{\frac{1}{4}}}$ – 2$x^{\frac{1}{8}}$ × $\frac{1}{x^{\frac{1}{8}}}=0$ ⇒ ($x^{\frac{1}{8}} – \frac{1}{x^{\frac{1}{8}}})^2=0$ ⇒ $x^{\frac{1}{8}}$ – $\frac{1}{x^{\frac{1}{8}}} = 0$ ⇒ $x^{\frac{1}{8}} = \frac{1}{x^\frac{1}{8}}$ ⇒ $x^{\frac{1}{4}} = 1$ ⇒ $x$ = 1 Now, $x^{81}+ \frac{1}{x^{81}}$= (1) 81 + $\frac{1}{(1)^{81}} = 2$ Hence, the correct answer is 2.
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