Question

How do you express 396 as a product of its prime factors, giving your answer in index notation?

Answer 1

Hint: To find the prime factors of the given number we will precede using prime factorization method. Starting with the smallest prime number we represent the number 396 as a product of prime factors such as when we multiplied the numbers we will get back the same number.

Complete step by step solution:
We have been given a number 396.
We have to express the number as a product of prime factors.
By using the prime factorization method we will represent the number as a product of prime numbers.
The number 396 is represented as a product of prime factors as
\[\begin{align}
  & 2\left| \!{\underline {\,
  396 \,}} \right. \\
 & 2\left| \!{\underline {\,
  198 \,}} \right. \\
 & 3\left| \!{\underline {\,
  99 \,}} \right. \\
 & 3\left| \!{\underline {\,
  33 \,}} \right. \\
 & 11\left| \!{\underline {\,
  11 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
$\Rightarrow 396=2\times 2\times 3\times 3\times 11$
Here in the above representation all numbers in the product are prime numbers which leaves remainder zero when 396 is divided by any one of them.
Hence above is the required prime factor representation of the number 396.

Note: The point to be noted is that here in this particular question we have to use only prime numbers to express the number as a product. Prime numbers are those numbers which have only two factors i.e. the number itself and 1. Prime numbers are always greater than 1. 2 is the smallest prime number. Prime factorization method is also useful to find the square root and cube roots of any number.