Question

is 3240 a perfect cube? if not then by which smallest number should 3240 be multiplied so that the product is a perfect cube?​

Answer 1

3240 it's not a perfect cube.

The smallest number should 3240 be multiplied so that the product is a perfect cube is 15² = 225.

Explanation:

3240 is a perfect cube if 3240 = n³ (n ∈ N).

Use the Prime Factorization:

`\begin{array}c3240&2\\1620&2\\810&2\\405&5\\81&3\\27&3\\9&3\\3&3\\1\end{array}\\\\3240=2\cdot2\cdot2\cdot5\cdot3\cdot3\cdot3\cdot3=2^3\cdot3^3\cdot5\cdot3=(2\cdot3)^3\cdot5\cdot3=6^3\cdot5\cdot3`

`3240=6^3\cdot15\qquad\text{multiply both sides by}\ 15^2\\\\3240\cdot15^2=6^3\cdot15^3=3240\cdot15^2=(6\cdot15)^3=90^3`

Used:

`a^n\cdot a^m=a^(n+m)\\\\(ab)^n=a^nb^m`