Question

if x = 4+root 15, y = 4-root 15, find x^3 + y^3

Answer 1

x
${x}^{3} + y {3} = (x + y)(x ^{2} + {y}^{2} - x \times y) \\ = (4 + \sqrt{15 } + 4 - \sqrt{15} )((4 + \sqrt{15} )^{2} + (4 - \sqrt{15} ) ^{2} - (4 + \sqrt{15}) (4 - \sqrt{15} )) \\ = (8)(16 + 15 + 8 \sqrt{15} + 16 + 15 - 8 \sqrt{15} - (4 ^{2} - ( \sqrt{15})^{2} ) \\ = (8)(31 + 31 - 1) \\ =(8)(61) \\ = 488$

Answer 2

Use the identity.

X³+Y³ = (X+Y)(X²+XY+Y²)

= (4+√15+4-√15)( (4+√15)²+(4+√15)(4-√15)+ (4-√15)²

= 8(16+15+16+15+16-15)
= 8*61 = 488