Question

How do you write the prime factorization of $360$ in exponential form.

Answer 1

Hint: Within the problem we will observe the term ‘prime factorization’. We all know that prime factorization is the process of factoring a number with only using the prime numbers starting with $2$. So first we will check whether the given number is divisible by $2$ or not. If yes then we will write the given number as a product of $2$ and the quotient. Again, we are going to factorize the obtained quotient by following the above process. We will continue with the above process until we will get the quotient as $1$. While doing the above process if you get the any number is not divisible by $2$, then check for the remaining prime numbers $3$, $5$, $7$, $11...$.

Formulas Used:
1. $a\times a\times a\times a\times a....\text{n times}={{a}^{n}}$.

Complete Step by Step Procedure:
Given number $360$.
Checking whether the number $360$ is divisible by $2$ or not. We can observe that the number $360$ is divisible by $2$ and the quotient will be $180$. Then we can write the number $360$ as
$360=2\times 180$
Now checking if the number $180$ is divisible by $2$ or not. We can observe that the number $180$ is divisible by $2$ and the quotient will be $90$. Then we can write the number $180$ as
$180=2\times 90$
From the above value we can write $360$ as
$360=2\times 2\times 90$
Now checking if the number $90$ is divisible by $2$ or not. We can observe that the number $90$ is divisible by $2$ and the quotient will be $45$. Then we can write the number $90$ as
$90=2\times 45$
From the above value we can write $360$ as
$360=2\times 2\times 2\times 45$
Now checking if the number $45$ is divisible by $2$ or not. We can observe that the number $45$ is not divisible by $2$. So, we are checking with the next prime number $3$. We can observe that the number $45$ is divisible by $3$. When we divide the number $45$ with $3$, then we will get $15$ as quotient, then we can write $45$ as
$45=3\times 15$
From the above value we can write $360$ as
$360=2\times 2\times 2\times 3\times 15$
Now checking if the number $15$ is divisible by $2$ or not. We can observe that the number $15$ is not divisible by $2$. So, we are checking with the next prime number $3$. We can observe that the number $15$ is divisible by $3$. When we divide the number $15$ with $3$, then we will get $5$ as quotient, then we can write $15$ as
$15=3\times 5$
From the above value we can write $360$ as
$360=2\times 2\times 2\times 3\times 3\times 5$
Now we have the exponential rule $a\times a\times a\times a\times a....\text{n times}={{a}^{n}}$.

Applying this formula in the above value, then we will get
$360={{2}^{3}}\times {{3}^{2}}\times 5$

Note:
In the problem they have mentioned prime factorization so we have checked that the given number is divisible by prime numbers or not only. If they have only mentioned factorization, then we need to check whether the given number is divisible by all the numbers.