If a + b = 8 and $a + a^2b + b + ab^2 = 128$ then the positive value of $a^3 + b^3$ is :

152

a + b = 8 and a + a2 b + b + ab2 = 128

We know that,

a³ + b³ = (a + b) [(a + b)² - 3ab]

So,

a + a² b + b + ab² = 128

a + a² b + b + ab²  = 128

= 8 + a2 b + ab2 = 128

= a² b + ab² = 128 - 8

= a² b + ab² = 120

= ab (a + b) = 120

= ab × 8 = 120

= ab = 15

a³ + b³ = (a + b) [(a + b)² - 3ab]

= a³ + b³ = 8 [8² - 3 × 15]

= a³ + b³ = 8 [64 - 45]

= a³ + b³ = 8 × 19

a³ + b³ = 152