If a - b = 4 and $a^3 - b^3 = 88 $, then find the value of $a^2 - b^2$.

$8\sqrt{6}$

Given that,

a - b = 4

a³ - b³ = 88

We know that,

(a³ - b³) = (a - b) (a² + b² + ab)

(a³ - b³) = (a - b) [(a - b)² + 3ab]

(a + b)² = a² + b² + 2ab

(a + b)(a - b) = a² - b²

According to the given data,

88 = 4 × (4² + 3ab)

= 22 = 16 + 3ab

= 3ab = 22 - 16

= ab = 2

(a - b)² = a² + b² - 2ab

= 4² = a² + b² - 2 × 2

= a² + b² = 16 + 4 

= a² + b² = 20

(a + b)² = 20 + 2 × 2

= a + b = √24 = 2√6

Then (a + b)(a - b) = 2√6 × 4 = $8\sqrt{6}$