Question

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

Answer 1

Hint: Cube root can be defined as the number which produces a given number when cubed. In other words, the cube-root of a number is the factor which is multiplied by it three times to get that number. It is expressed as $ \sqrt[3] {n} $ and is read as the cube root of the number “n”. For example $ \sqrt[3] {{27}} = \sqrt[3] {{3 \times 3 \times 3}} = 3 $ since here, the number “ $ 3 $ “ is repeated three times. To find cube-root, first we will find the prime factors by division method.

Complete step-by-step answer:
Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than $ 1 $ and which are not the product of any two smaller natural numbers.
To find the prime factorisation, start dividing the given number $ 13824 $ by the first prime number, $ 2 $ Continue division until it is not divisible by $ 2 $ . Gradually start dividing by $ 3,{\text{ 5, 7, 11, 13,}}.... $
 \[\Rightarrow 13824 = \underline {2 \times 2 \times 2} \times \underline {2 \times 2 \times 2} \times \underline {2 \times 2 \times 2} \times \underline {3 \times 3 \times 3} \]
When the same number is multiplied thrice, it can be written as a cube of the number.
 \[\Rightarrow 13824 = {2^3} \times {2^3} \times {2^3} \times {3^3}\]
Take cube root on both the sides of the equation –
 \[\Rightarrow \sqrt[3] {{13824}} = \sqrt[3] {{{2^3} \times {2^3} \times {2^3} \times {3^3}}}\]
Cube and cube-root cancel each other on the right side of the equation.
 \[\Rightarrow \sqrt[3] {{13824}} = 2 \times 2 \times 2 \times 3\]
Simplification-
 \[\sqrt[3] {{13824}} = 24\]
So, the correct answer is “Option A”.

Note: You should be very good in multiples. As, ultimately your answer depend on the multiplication of the numbers. Also, know the basic difference between the cubes and cube roots and apply accordingly.