Question

If $x+\frac{1}{x}=5$, find the value of $x^4+\frac{1}{x^4}$.

• A. $144$
• B. $400$
• C. $236$
• D. $527$

Answer 1

Answer: (D)

$527$

Given $x+\frac{1}{x}=5$

let $y=x^4+\frac{1}{x^4}$

$x+\frac{1}{x}=5$

Squaring on both sides

$(x+\frac{1}{x}{)}^2=5^2$

$\Rightarrow x^2+\frac{1}{x^2}+2\left(x\right)\left(\frac{1}{x}\right)=25$

$\Rightarrow x^2+\frac{1}{x^2}+2=25$

$\Rightarrow x^2+\frac{1}{x^2}=23$

Squaring on both sides

$\Rightarrow (x^2+\frac{1}{x^2}{)}^2={23}^2$

$\Rightarrow x^4+\frac{1}{x^4})+2(x^2\left)\right(\frac{1}{x^2})=529$

$\Rightarrow x^4+\frac{1}{x^4}+2=529$

$\Rightarrow x^4+\frac{1}{x^4}=527$

$\Rightarrow y=527$

Thus, required term value is $527$.