Question

by what smallest number should $10584$ be multiplied so that product may be a perfect square ? also find the square root of the new number.

Answer 1

Prime factorisation of $10584$ is

$10584=2^3\times 3^2\times 7^2$

Now for the above to be a perfect square we can see that we need to multiply $2\times 3=6$

$\therefore$ the least number to be multiplied to $10584$ to make it a perfect square is $6$

$\therefore 10584\times 6=2^3\times 3^3\times 7^2\times 2\times 3=2^4\times 3^4\times 7^2$

The square root of this new number is

$\sqrt{2^4\times 3^4\times 7^2}=2^2\times 3^2\times 7=4\times 9\times 7=63\times 4=192$