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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Question : If $x-\frac{1}{x}=1$, then what is the value of $\left (\frac{1}{x-1}-\frac{1}{x+1}+\frac{1}{x^{2}+1}-\frac{1}{x^{2}-1} \right)\;$?
Question : If $\left(x^2+\frac{1}{x^2}\right)=7$, and $0<x<1$, find the value of $x^2-\frac{1}{x^2}$.
Question : If $x=\frac{\sqrt{5}+1}{\sqrt{5}-1}$ and $y=\frac{\sqrt{5}-1}{\sqrt{5}+1}$, then the value of $\frac{x^{2}+xy+y^{2}}{x^{2}-xy+y^{2}}$ is:
Question : If $2x+\frac{1}{2x}=2,$ what is the value of $\sqrt{2\left (\frac{1}{x}\right)^{4}+\left (\frac{1}{x}\right)^{5}}\; ?$
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