If x3 + y3 = 637 and x + y = 13, then find the value of x4 + y4.

Short trick?

Take x = 8 and y = 5

⇒ x4 + y4 = 84 + 54

⇒ 4096 + 625 = 4721

Detailed solution?

Given that x3 + y3 = 637 and x + y = 13;

We know that?

x3 + y3 = (x + y) (x2 + y2 – xy)

⇒ x3 + y3 = (x + y) ((x + y)2 – 3xy)

⇒ 647 = 13 × (169 – 3xy)

⇒ 49 = 169 – 3xy

⇒ 3xy = 120

⇒ xy = 40

x4 + y4 = (x2 + y2)2 – 2x2y2

⇒ ((x + y)2 – 2xy)2 – 2x2y2

⇒ (169 – 80)2 – 2 × 1600

⇒ 892 – 3200

⇒ 4721