How do you write the prime factorization of $ 392 $ .

Hint: In the problem, we can observe the term ‘prime factorization’. We know that prime factorization is the process of factorizing 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 will 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 any number that is not divisible by $ 2 $, then check for the remaining prime numbers $ 3 $, $ 5 $, $ 7 $, $ 11... $.

Complete step by step answer:
Given number $ 392 $ .
Checking whether the number $ 392 $ is divisible by $ 2 $ or not. We can observe that the number $ 392 $ is divisible by $ 2 $ and the quotient will be $ 196 $ . Then we can write the number $ 392 $ as
 $ 392=2\times 196 $
Now checking if the number $ 196 $ is divisible by $ 2 $ or not. We can observe that the number $ 196 $ is divisible by $ 2 $ and the quotient will be $ 98 $ . Then we can write the number $ 196 $ as
 $ 196=2\times 98 $
From the above value we can write $ 392 $ as
 $ 392=2\times 2\times 98 $
Now checking if the number $ 98 $ is divisible by $ 2 $ or not. We can observe that the number $ 98 $ is divisible by $ 2 $ and the quotient will be $ 49 $ . Then we can write the number $ 98 $ as
 $ 98=2\times 49 $
From the above value we can write $ 392 $ as
 $ 392=2\times 2\times 2\times 49 $
Now checking the number $ 49 $ is divisible by $ 2 $ or not. We can observe that the number $ 49 $ is not divisible by $ 2 $ . So, we are checking with the next prime number $ 3 $ . We can observe that the number $ 49 $ is not divisible by $ 3 $ . Now checking with the next prime number $ 5 $ . Here also the number $ 49 $ is not divisible by $ 5 $ . Now checking with the next prime number $ 7 $ . We can observe that the number $ 49 $ is divisible by $ 7 $ and the quotient will be $ 7 $ , then we can write the number $ 49 $ as
 $ 49=7\times 7 $
From the above value we can write $ 392 $ as
 $ 392=2\times 2\times 2\times 7\times 7 $
Applying the exponential rule $ a.a.a.a.a.a...\text{ n times}={{a}^{n}} $ in the above equation, then we will have
$\Rightarrow$ $ 392={{2}^{3}}\times {{7}^{2}} $ .

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.