The sum of two numbers is 9 and their product is 20. Find the sum of their :

(i) squares

(ii) cubes.

Let two numbers be a and b.

Given,

Sum of numbers = 9 and Product = 20.

∴ a + b = 9 and ab = 20.

(i) Sum of squares = a² + b²

By formula,

⇒ (a + b)² = a² + b² + 2ab

Substituting values we get :

⇒ 9² = a² + b² + 2 × 20

⇒ 81 = a² + b² + 40

⇒ a² + b² = 81 - 40 = 41.

Hence, sum of squares = 41.

(ii) Sum of cubes = a³ + b³

By formula,

⇒ (a + b)³ = a³ + b³ + 3ab(a + b)

⇒ 9³ = a³ + b³ + 3 × 20 × 9

⇒ 729 = a³ + b³ + 540

⇒ a³ + b³ = 729 - 540

⇒ a³ + b³ = 189.

Hence, a³ + b³ = 189.