Question

Find the cube root of the following numbers by prime factorization.
(a) 110592 (b) 54872

Answer 1

Hint: In the process of prime factorization, we will first factorize the given numbers into prime factors. We will start with the smallest prime factor 2 and keep continuing the process with 3, 5, 7 … if there is a need. After obtaining all the prime factors, we will group the prime factors in sets of three. We will put three same prime factors together and then take one prime factor from each set and multiply them to obtain the cube root.

Complete step-by-step solution:
In order to find the cube root of a number by prime factorization method, we use the following steps:
First of all, we resolve the given number into its prime factors.
Then, we group the factors in three in such a way that each number of the group is the same.
Take one factor from each group and then find the product of the factors obtained in the previous step. This product is the required cube root of the number.
Since the first number given to us is 110592.
We will first resolve this number into its prime factors.
We know that a prime number is one, which has only two factors, that is, 1 and itself. It means that a number that is divisible by only 1 and itself is a prime number.
So, we can write the given number as:
$\begin{align}
  & 110592=2\times 2\times 2\times 2\times 3\times 3\times 3\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 \\
 & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left( 2\times 2\times 2 \right)\times \left( 2\times 2\times 2 \right)\times \left( 2\times 2\times 2 \right)\times \left( 2\times 2\times 2 \right)\times \left( 3\times 3\times 3 \right) \\
\end{align}$
So, after making a group of three, we take out one factor from each group and multiply it. Therefore, cube root of 110592 is = $2\times 2\times 2\times 2\times 3=48$
Similarly for the second number which is 54872. On resolving it into its prime factors, we get:
$54872=\left( 2\times 2\times 2 \right)\times \left( 19\times 19\times 19 \right)$
After making group of three and taking one factor from each group and on multiplying them, we get cube root of 54872 as = $2\times 19=38$
Hence, the cube root of 110952 is 48 and of 54872 is 38.

Note: Students should note here that while doing prime factorization, it is not necessary that we start from the smallest prime number that divides the given number. We can divide it by any number provided that the number must be a prime.