If $a^3 + b^3 = 20$ and a + b = 5, then find the value of $a^4+b^4$.

23

a + b = 5

a³ + b³ = 20

 (a + b)³ = a³ + b³ + 3ab(a + b)

 (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

We know that, (a² + b²)² = a⁴ + b⁴ + 2a²b²

11² = a⁴ + b⁴ + 2 × 49

= a⁴ + b⁴ = 121 – 98 = 23