Question

Find the smallest number by which each of the following number must be multiplied to obtain a perfect cube. a. 1296

Answer 1

Let us first split the number 1296 in its prime factors
So,
1296=2×2×2×2×3×3×3×3
So,
For this number to be a perfect cube the prime factors must be of the number which is divisible by 3

So we can see that there are four 3s and four 2s in the number 1296 prime factors
So, to make this perfect cube, there should be more two 3s and two 2s in the multiplications of the new formed number(Six 3s and Six 2s)
This is because 6 is the least number greater than 4 which is divisible by 3
So,
New number= 2×2×2×2×2×2×3×3×3×3×3×3
                     =46656 (Which is a perfect cube)
So, the least number that can be multiplied to make 1296 a perfect cube is 
2 ×2×3×3
=36
So, 36 is the required number

Answer 2

Given number is 1296.

Let us first see that 1296 is the Perfect cube or not. 

First break this number into the smaller numbers which do not hvae any other factors except 1. 

∴ 1296 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 
Since, there is the requirement of the 2. 

∴ 1296 × 2 × 2 × 3 × 3 (or 46656) will equal to 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

Now, 46656 will be the new number, which can form the perfect cubes. 


Hence, The perfect cube is 2 × 2 × 3 × 3 =  4 × 12 = 36.


Hope it helps.