Question : If $\left(x+\frac{1}{x}\right)=5 \sqrt{2}$, then what is the value of $\left(x^4+x^{-4}\right)$?
Option 1: 2542
Option 2: 2650
Option 3: 2452
Option 4: 2302
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Correct Answer: 2302
Solution :
Given: $\left(x+\frac{1}{x}\right)=5 \sqrt{2}$
Squaring both sides, we get,
⇒ $(x+\frac{1}{x})^2=(5 \sqrt{2})^2$
⇒ $x^2+\frac{1}{x^2}+2=50$
⇒ $x^2+\frac{1}{x^2}=48$
Again squaring both sides, we get:
⇒ $(x^2+\frac{1}{x^2})^2=48^2$
⇒ $x^4+\frac{1}{x^4}+2=2304$
$\therefore x^4+\frac{1}{x^4}=2302$
Hence, the correct answer is 2302.
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