Difference of two perfect cubes is 387. If the cube root of the greater of the two numbers is 8, find the cube root of the smaller number.

Cube root of greater number = 8.

Let the cube root of smaller number be $x$.

Hence,

$8^3 - x^3 = 387\\[1em] ⇒ 512 - x^3 = 387\\[1em] ⇒ 512 - 387 = x^3\\[1em] ⇒ 125 = x^3\\[1em] ⇒ x = \sqrt[3]{125}\\[1em] ⇒ x = \sqrt[3]{5\times 5\times 5}\\[1em] ⇒ x = 5$

Hence, the cube root of the smaller number is 5