If (X + (1 / X)) = 4, then the value of \(X^{4} + (1 / X^{4})\) is (a) 124 (b) 64 (c) 194 (d) Can’t be determined

Solution

Given \((X + \frac{1}{X}) = 4\), we need to find the value of \(X^{4} + \frac{1}{X^{4}}\).

Step 1: Square \((X + \frac{1}{X})\)

First, square both sides to find \(X^2 + \frac{1}{X^2}\):

\[
(X + \frac{1}{X})^2 = 4^2
\]
\[
X^2 + 2 + \frac{1}{X^2} = 16
\]
\[
X^2 + \frac{1}{X^2} = 16 – 2
\]
\[
X^2 + \frac{1}{X^2} = 14
\]

Step 2: Square \(X^2 + \frac{1}{X^2}\)

Next, square both sides again to find \(X^4 + \frac{1}{X^4}\):

\[
(X^2 + \frac{1}{X^2})^2 = 14^2
\]
\[
X^4 + 2 + \frac{1}{X^4} = 196
\]
\[
X^4 + \frac{1}{X^4} = 196 – 2
\]
\[
X^4 + \frac{1}{X^4} = 194
\]

Therefore, the value of \(X^4 + \frac{1}{X^4}\) is 194.

The correct answer is (c) 194.