If \(a + \frac{1}{a} = 5 \) , then what is the value of \({a^3} + \frac{1}{{{a^3}}}\)?

Question

Question

This question was previously asked in

SSC CGL 2022 Tier-I Official Paper (Held On : 12 Dec 2022 Shift 2)

Answer 1

Given:

\(a + \frac{1}{a} = 5 \)

Concept used:

(a + b)3 = a3 + 3ab(a + b) + b3

Calculation:

\(a + \frac{1}{a} = 5 \)

⇒ \((a + \frac{1}{a})^3 = 5^3 \)

⇒ \(a^3 + \frac{1}{a^{3}} + 3(a + \frac {1}{a}) = 5^3\)

⇒ \(a^3 + \frac{1}{a^{3}} + 3 \times 5 = 125\)

⇒ \(a^3 + \frac{1}{a^{3}} = 125 - 15\)

⇒ \(a^3 + \frac{1}{a^{3}} = 110 \)

∴ The value of \({a^3} + \frac{1}{{{a^3}}}\) is 110.