Question

Find the cube root of the number 91125 by prime factorization method.

Answer 1

Hint: First reduce the given number as the multiple of powers of its prime factors. This will be in the form of $N = {p^a} \times {q^b} \times {r^c} \times ....$, where \[p,{\text{ }}q,{\text{ }}r,...\] are prime factors of $N$. Then for finding the cube root of $N$, divide the powers of prime numbers by 3. The number left after dividing the powers by 3 will be the answer.

Complete step-by-step answer:
According to the question, the given number is 91125. Its cube root is to be determined using the prime factorization method.
On applying the prime factorization method, first we will reduce the number as the multiple of powers of its prime factors.
As we can clearly see that the number 91125 is divisible by 5, which is a prime number. So we can write it as:
$ \Rightarrow 91125 = 5 \times 18225$
The number left i.e. 18825 can also be divided by 5 repeatedly. Doing so, we’ll get:
\[
   \Rightarrow 91125 = 5 \times 5 \times 3645 \\
   \Rightarrow 91125 = 5 \times 5 \times 5 \times 729
 \]
We know that 729 can be written as 3 raise to the power 6 i.e. ${3^6}$. Doing this and writing 5’s also in powers, we’ll get:
$ \Rightarrow 91125 = {5^3} \times {3^6}$
Now, for finding the cube root of 91125, we’ll divide the powers by 3 on both sides. So we have:
$
   \Rightarrow {\left( {91125} \right)^{\dfrac{1}{3}}} = {5^{\dfrac{3}{3}}} \times {3^{\dfrac{6}{3}}} \\
   \Rightarrow {\left( {91125} \right)^{\dfrac{1}{3}}} = 5 \times {3^2} = 5 \times 9 \\
   \Rightarrow {\left( {91125} \right)^{\dfrac{1}{3}}} = 45
 $

Thus the cube root of the number 91125 is 45.

Note: In case we have to find the square root of any number by the same method, we follow the same steps. First we reduce the given number as the multiple of powers of its prime factors. And in the last step, we divide the powers by 2 instead of 3. Similar is the case with other roots. If the fourth root is required, divide the powers by 4 in the last step and so on.