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.

Explanation

By factorization,
`8192=2xx2xx2xx2xx2xx2xx2xx2xx2xx2xx2`
`8192=2^3xx2^3xx2^3xx2`
Since, one triples remain incomplete, hence `8192` is not a perfect cube.
So, we divide `8192` by `2` to make its quotient a perfect cube.
Since, `8192/2=4096`
`4096=2xx2xx2xx2xx2xx2xx2xx2xx2xx2xx2xx2`
`4096 =2^3times2^3xx2^3xx2^3`
Cube root of `4096`,
`root(3)(4096)=root(3)(2^3xx2^3xx2^3xx2^3)`
`root(3)(4096)=2xx2xx2xx2=16`