If \(\left(x+\frac{1}{x}\right)=5 \sqrt{2}\), then what is the value of (x⁴ + x^{- 4})?

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 01 Dec 2022 Shift 2)

Answer 1

Given:

\(\left(x+\frac{1}{x}\right)=5 \sqrt{2}\)

Concept used:

\((a + \frac {1}{a})^2 = a^2 + \frac {1}{a^2} + 2\)

Calculation:

\(\left(x+\frac{1}{x}\right)=5 \sqrt{2}\)

⇒ \(\left(x+\frac{1}{x}\right)^2={(5 \sqrt{2})}^2\)

⇒ \(x^2 + \frac {1}{x^2} + 2 = 50\)

⇒ \({(x^2 + \frac {1}{x^2})^2} = 48^2\)

⇒ \({(x^4 + \frac {1}{x^4}+ 2)} = 2304\)

⇒ \({(x^4 + \frac {1}{x^4})} = 2302\)

∴ The required value of \({(x^4 + \frac {1}{x^4})}\) or (x4 + x- 4) is 2302.