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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