Question

Find the cube root of the following number by prime factorisation method: $21952$

Answer 1

Hint: We will start by defining prime numbers and factors and then we will show what is the prime factorisation method. Then we will take the given number in the question and start dividing it by the smallest numbers such that the remainder is zero, we will continue doing so until we find all the factors then at last to find the cube root we will just take 1 number out of 3 same numbers and we will get our cube root.

Complete step by step answer:
First let’s understand what prime factorisation is, to understand the prime factorisation method let's define what are prime numbers?
So basically, a prime number is a whole number greater than 1 that cannot be made by multiplying other whole numbers. The first few prime numbers are : $2,3,5,7,11,13......$
Let’s now see what is meant by factors? Factors are the numbers you multiply together to get another number. For example: $2\times 3=6$ , here $2,3$ are the factors of $6$
 Now let’s finally see the definition of prime factorisation. Prime factorisation is a method of finding the prime numbers which multiply together to make the original number.
For example: What are the prime factors of $12$?
It is best to start working from the smallest prime number, which is $2$ , so let's check: $\dfrac{12}{2}=6$. It is divided exactly by $2$. But since $6$ is not a prime number, so we need to go further. Let's try 2 again: $\dfrac{6}{2}=3$ . And 3 is a prime number, so we have the answer: $12=2\times 2\times 3$.
 As you can see, every factor is a prime number, so the answer must be right.
Similarly, we have to find out the cube root for the given number that is: $21952$. First let’s find the factors of this number:
First, let’s take the smallest prime number i.e. $2$: $\dfrac{21952}{2}=10976$ , since it is divided exactly by $2$ , we have $2$ as one of the factors.
Moving further, let ‘s keep dividing it by 2 :
 $\begin{align}
  & \dfrac{10976}{2}=5488\Rightarrow 2 \\
 & \dfrac{5488}{2}=2744\Rightarrow 2 \\
 & \dfrac{2744}{2}=1372\Rightarrow 2 \\
 & \dfrac{1372}{2}=686\Rightarrow 2 \\
 & \dfrac{686}{2}=343\Rightarrow 2 \\
\end{align}$
Now, let’s try a higher prime number that is $7$ :
\[\begin{align}
  & \dfrac{343}{7}=49\Rightarrow 7 \\
 & \dfrac{49}{7}=7\Rightarrow 7 \\
 & \dfrac{7}{7}=1\Rightarrow 7 \\
\end{align}\]
Since it can’t be divided further, we got all the prime factors of the given number that is: $21952$
Therefore: $21952=2\times 2\times 2\times 2\times 2\times 2\times 7\times 7\times 7$
For finding cube root, we will make a group of 3 same prime factors:
$\begin{align}
  & 21952={{\left( 2 \right)}^{3}}\times {{\left( 2 \right)}^{3}}\times {{\left( 7 \right)}^{3}} \\
 & 21952={{\left( 28 \right)}^{3}} \\
 & \sqrt[3]{21952}=28 \\
\end{align}$

Therefore, the cube root of $21952$ is $28$.

Note: A prime number can only be divided by 1 or itself, so it cannot be factored any further. Every other whole number can be broken down into prime number factors. Prime Numbers are the basic building blocks of all numbers. When finding the cube roots make sure you have a group of three same prime factors similarly for square root we need a group of two prime factors.