Question : if $x+\frac{1}{x}=2$, then the value of $x^4+\frac{1}{x^4}$=__________.
Option 1: 0
Option 2: 2
Option 3: –1
Option 4: 1
Correct Answer: 2
Solution :
Given: $x+\frac{1}{x}=2$
Squaring both sides, we get
⇒ $(x+\frac{1}{x})^2=2^2$
⇒ $x^2+\frac{1}{x^2}+2\times x\times\frac{1}{x}=4$
⇒ $x^2+\frac{1}{x^2}=4-2$
⇒ $x^2+\frac{1}{x^2}=2$
Again squaring both sides, we get:
⇒ $(x^2+\frac{1}{x^2})^2=2^2$
⇒ $x^4+\frac{1}{x^4}+2\times x^2\times\frac{1}{x^2}=4$
⇒ $x^4+\frac{1}{x^4}=4-2$
⇒ $x^4+\frac{1}{x^4}=2$
Hence, the correct answer is 2.
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