Question

If \(a^3 + b^3 = 20\) and a + b = 5, then find the value of a⁴ + b⁴
1. 26
2. 23
3. 25
4. 24

Answer 1

Correct Answer - Option 2 : 23

Given :

a + b = 5 and a³ + b³ = 20

Formula used :

(a + b)³ = a³ + b³ + 3ab(a + b)

(a² + b²)² = a⁴ + b⁴ + 2a²b²

Calculations :

(a + b)³ = a³ + b³ + 3ab(a + b)

5³ = a³ + b³ + 3ab(5)

⇒ 3ab(5) = 125 – 20

⇒ ab = 7

Now,

(a + b)² = a² + b² + 2ab

⇒ 5² =  a² + b² + 2 × 7

⇒ a² + b² = 25 – 14 = 11

So,

(a² + b²)² = a⁴ + b⁴ + 2a²b²

11² = a⁴ + b⁴ + 2 × 49

⇒ a⁴ + b⁴ = 121 – 98

⇒ 23

∴ The value of a⁴ + b⁴ is equal to 23.