Question

Find the cube root of 46656 by prime factorization method.

Answer 1

Hint: In this question we have to find the cube root of the given number by prime factorization.
We have to find out the cube root of 46656 using the prime factorization method. Prime numbers refers to those numbers which are divisible by one and itself, so find out all the factors of the given number and then look only for prime factors amongst the factors. Cube root of any number P means ${\left( P \right)^{\dfrac{1}{3}}}$.

Complete step-by-step answer:

$ \Rightarrow \sqrt[3]{{46656}} = {\left( {46656} \right)^{\dfrac{1}{3}}}$……………………. (1)
So, first factorize the given number.

Therefore 46656 factorization \[{\text{ = 2}} \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\], (2 and 3 is a prime number)

A number is called a prime number if the factor of the number is either 1 or itself.
So, 2 is multiplied six times and 3 is also multiplied six times together to make the original number.
$\therefore $Prime factorization of 46656\[ = {2^6} \times {3^6}\]

So substitute this value in above equation we have,
$ \Rightarrow \sqrt[3]{{46656}} = {\left( {46656} \right)^{\dfrac{1}{3}}} = {\left( {{2^6} \times {3^6}} \right)^{\dfrac{1}{3}}}$

Now simplify the above equation we have,
$ \Rightarrow \sqrt[3]{{46656}} = {\left( {{2^6} \times {3^6}} \right)^{\dfrac{1}{3}}} = {2^{6 \times \dfrac{1}{3}}} \times {3^{6 \times \dfrac{1}{3}}} = {2^2} \times {3^2} = 4 \times 9 = 36$
So this is the required cube root using the prime factorization method.

Note: Whenever we face such types of problems the key concept is to have the basic understanding of the factors which are prime. This can be understood by having the gist of definition of prime factors. This concept will help you get on the right track to reach the answer.