The number $2^p \times 5^q \times \dfrac{5}{2}$ is a perfect cube where $p$ and $q$ are greater than 1. Find the smallest possible integer values of $p$ and $q$. Answer p = ... q = ...

1. Rewrite the number using laws of indices

$2^p \times 5^q \times \frac{5}{2}=2^p \times 5^q \times 5 \times 2^{-1}=2^{p-1}\times 5^{q+1}$

2. Make each prime exponent a multiple of 3 and choose the smallest allowed values

For $2^{p-1}\times 5^{q+1}$ to be a perfect cube, both exponents must be multiples of $3$.

So
$p-1=3,6,9,\ldots$
Since $p>1$, the smallest possible value is
$p=4.$
Also
$q+1=3,6,9,\ldots$
Since $q>1$, the smallest possible value is
$q=2.$