Question

If $x+\frac{1}{x}=4$, then $x^4+\frac{1}{x^4}$ is equal to

• A. $196$
• B. $194$
• C. $192$
• D. $190$

Answer 1

The correct option is B

$194$

Explanation for the correct option,

It is given that, $x+\frac{1}{x}=4$.

On squaring both sides, we get;

$$\begin{gathered} {\left(x+\frac{1}{x}\right)}^2=16 \\ \Rightarrow x^2+\frac{1}{x^2}+2=16 \\ \Rightarrow x^2+\frac{1}{x^2}=14 \end{gathered}$$ [Since, ${(a+b)}^2=a^2+b^2+2ab$]

Again squaring on both sides, we get;

$$\begin{gathered} {\left(x^2+\frac{1}{x^2}\right)}^2=196 \\ \Rightarrow x^4+\frac{1}{x^4}+2=196 \\ \Rightarrow x^4+\frac{1}{x^4}=194 \end{gathered}$$

Therefore, the value of $x^4+\frac{1}{x^4}$ is $194$.

Hence, option $B$ is the correct option.