Question

Find the cube root of 5832 by the prime factorization method.

Answer 1

Hint: Prime factorization method is a technique of expressing a number using its prime factors. Prime numbers are $2,3,5,7,11....$so on. Start with dividing the number $5832$ with the smallest possible prime factor, i.e. $2$ . Repeat this process until we reach the number $1$ . Now using exponents, express the number in the form of the product of its prime factors. Now use $\sqrt[3]{m} = {\left( m \right)^{\dfrac{1}{3}}}$ to find the cube root of the number.

Complete step-by-step answer:
Here in this problem we are given a positive integer $5832$ and using the prime factorization method, we need to find the cube root of this number
Before starting with the solution we must understand the principles of the prime factorization method. Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers. These prime numbers when multiplied with any natural numbers produce composite numbers.
For example, the prime factors of $126$ will be $2,3{\text{ and }}7$ as $2 \times 3 \times 3 \times 7 = 126$ and $2,3{\text{ and }}7$ are prime numbers.
For the given number $5832$ we can find the prime factors step by step dividing the number by the smallest prime number possible as:

Steps	Prime Factor
Step $1$ : Dividing by $2$	$2$	\[5832 \div 2 = 2916\]
Step $2$ : Dividing by $2$	$2$	$2916 \div 2 = 1458$
Step $3$ : Dividing by $2$	$2$	$1458 \div 2 = 729$
Step $4$ : Dividing by $3$	$3$	$729 \div 3 = 243$
Step $5$ : Dividing by $3$	$3$	$243 \div 3 = 81$
Step $6$ : Dividing by $3$	$3$	$81 \div 3 = 27$
Step $7$ : Dividing by $3$	$3$	$27 \div 3 = 9$
Step $8$ : Dividing by $3$	$3$	$9 \div 3 = 3$
Step $9$ : Dividing by $3$	$3$	$3 \div 3 = 1$

Therefore, the prime factors of $5832$ are $2{\text{ and }}3$; and the product can be expressed as $5832 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = {2^3} \times {3^6}$
We know that $\sqrt[3]{m} = {\left( m \right)^{\dfrac{1}{3}}}$
So the cube root of $5832$ will be expressed as $\sqrt[3]{{5832}}$ which is also written as ${\left( {5832} \right)^{\dfrac{1}{3}}}$
Therefore, we get:
$ \Rightarrow {\left( {5832} \right)^{\dfrac{1}{3}}} = {\left( {{2^3} \times {3^6}} \right)^{\dfrac{1}{3}}}$
Since we know that the exponents are distributive over the multiplication, we get:
$ \Rightarrow {\left( {5832} \right)^{\dfrac{1}{3}}} = {\left( {{2^3} \times {3^6}} \right)^{\dfrac{1}{3}}} = {2^{3 \times \dfrac{1}{3}}} \times {3^{6 \times \dfrac{1}{3}}}$
Now the expression in the powers can be easily solved as:
\[ \Rightarrow {\left( {5832} \right)^{\dfrac{1}{3}}} = {2^{3 \times \dfrac{1}{3}}} \times {3^{6 \times \dfrac{1}{3}}} = {2^1} \times {3^2}\]
Therefore, we get the value for cube root as $\sqrt[3]{{5832}} = 2 \times {3^2} = 2 \times 9 = 18$

Hence, the cube root of the given number $5832$ is $18$ .

Note: In questions like this the use of the method of prime factorization plays an important role. But this method will not be sufficient when the given number is not a perfect cube. For example, if we try to find the cube root of $1458$ using prime factorization, we will get ${\left( {1458} \right)^{\dfrac{1}{3}}} = {\left( {{2^2} \times {3^6}} \right)^{\dfrac{1}{3}}} = 9 \times \sqrt[3]{4}$ . This way the cube root cannot be obtained when the number is not a perfect cube using this method.