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: Determine \( x^2 + \frac{1}{x^2} \) using the identity. Given \( x + \frac{1}{x} = 4 \). Squaring both sides yields \( \left(x + \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} = 16 \). Therefore, \( x^2 + \frac{1}{x^2} = 16 - 2 = 14 \). Step 2: Determine \( x^4 + \frac{1}{x^4} \) using the identity. Squaring the result from Step 1: \( \left(x^2 + \frac{1}{x^2} \right)^2 = x^4 + 2 + \frac{1}{x^4} \). Substituting the value found: \( 14^2 = x^4 + \frac{1}{x^4} + 2 \), which simplifies to \( 196 = x^4 + \frac{1}{x^4} + 2 \). Consequently, \( x^4 + \frac{1}{x^4} = 196 - 2 = 194 \).