Difference of two perfect cubes is” 189.

Given different of two perfect cubes $=189$

and cube root of the smaller number $=3$

$\therefore$ Cube of smaller number $=(3)^{3}=27$

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

Then, cube of larger number $=x^{3}$

According to the question,

$x^{3}-27=189$

$\Rightarrow \quad x^{3}=189+27$

$\Rightarrow \quad x^{3}=216$

$\Rightarrow \quad x=\sqrt[3]{216}=\sqrt[3]{6 \times 6 \times 6}$

$\therefore$ $x=6$

Hence, the cube root of the larger number is 6 .