Question

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

• A. 194
• B. 1945
• C. 190
• D. 1940

Hint:

To evaluate powers like \( x^4 + \frac{1}{x^4} \), start with known identities: - \( (x + \frac{1}{x})^2 = x^2 + \frac{1}{x^2} + 2 \) - \( (x^2 + \frac{1}{x^2})^2 = x^4 + \frac{1}{x^4} + 2 \)

Answer 1

The Correct Option is A

Step 1: Use the identity to find \( x^2 + \frac{1}{x^2} \). We are given: \[ x + \frac{1}{x} = 4 \] Squaring both sides: \[ \left(x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} = 16 \Rightarrow x^2 + \frac{1}{x^2} = 16 - 2 = 14 \] Step 2: Use identity to find \( x^4 + \frac{1}{x^4} \). We now square again: \[ \left(x^2 + \frac{1}{x^2} \right)^2 = x^4 + 2 + \frac{1}{x^4} \Rightarrow 14^2 = x^4 + \frac{1}{x^4} + 2 \Rightarrow 196 = x^4 + \frac{1}{x^4} + 2 \Rightarrow x^4 + \frac{1}{x^4} = 196 - 2 = 194 \]