Question

Find the cube root of 13824 by the method of prime factorization.
$
  A.{\text{ 24}} \\
  B.{\text{ 18}} \\
  C.{\text{ 12}} \\
  D.{\text{ 36}} \\
 $

Answer 1

Hint- In order to solve such a question of finding cube root by prime factorization method, first find the prime factorization of the number and further make the triplet of factors.

Complete step-by-step solution -
Given number: 13824
In order to find the prime factorization of the number, we continuously divide the number with prime numbers.
Prime factorization of number 13824 is:
$13824 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$
Further we make the triplet of factors:
$13824 = \left( {2 \times 2 \times 2} \right) \times \left( {2 \times 2 \times 2} \right) \times \left( {2 \times 2 \times 2} \right) \times \left( {3 \times 3 \times 3} \right)$
Further simplifying the RHS and writing it in the form of a cube.
\[
   \Rightarrow 13824 = {2^3} \times {2^3} \times {2^3} \times {3^3} \\
   \Rightarrow 13824 = {2^9} \times {3^3} \\
 \]
Now we will proceed with finding the cube root
\[
   \Rightarrow {\left( {13824} \right)^{\dfrac{1}{3}}} = {\left( {{2^9} \times {3^3}} \right)^{\dfrac{1}{3}}} \\
   \Rightarrow {\left( {13824} \right)^{\dfrac{1}{3}}} = {\left[ {{{\left( {{2^3} \times 3} \right)}^3}} \right]^{\dfrac{1}{3}}} \\
   \Rightarrow {\left( {13824} \right)^{\dfrac{1}{3}}} = {\left( {{2^3} \times 3} \right)^{3 \times \dfrac{1}{3}}}{\text{ }}\left[ {\because {{\left( {{a^m}} \right)}^n} = {a^{m \times n}}} \right] \\
   \Rightarrow {\left( {13824} \right)^{\dfrac{1}{3}}} = {2^3} \times 3 \\
   \Rightarrow {\left( {13824} \right)^{\dfrac{1}{3}}} = 8 \times 3 = 24 \\
 \]
Hence, the cube root of 13824 is 24.
So, option A is the correct option.

Note- In mathematics, a cube root of a number x is a number y such that ${y^3} = x$ . All nonzero real numbers have exactly one real cube root. Prime Factorization is finding which prime numbers multiply together to make the original number. Prime factorization method is one of the basic and easiest ways of finding the cube root of any number.