Question

What is the smallest number by which 8192 must be divided so that quotient is a perfect cube?
Also, find the cube root of the quotient so obtained.

Answer 1

The prime factors of 8192 are
8192 = 2×2×2×2×2×2×2×2×2×2×2×2×2×2
= 2³×2³×2³×2³×2
Since, one 2 is not in a group of 3’s (as a triplet), hence 8192 is not a perfect cube.
So, divide 8192 by 2 to make its quotient a perfect cube.

$\frac{8192}{2}=4096$
4096 = 2×2×2×2×2×2×2×2×2×2×2×2
= 2³×2³×2³×2³
Cube root of 4096 = $\sqrt[3]{34096}$
$\sqrt[2]{2(2\times 2\times 2\times 2{)}^3}$
= 2×2×2×2
= 16
Cube root of 4096 = 16