Question

Express 216 as a product of power of prime factors.

Answer 1

Hint: Here we will use the prime factorization method. In the prime factorization method, we will find the prime factors of the given number such that when we multiply the prime factors we will get the original number.

Complete step-by-step answer:
First we will break 216 into smaller factors. For 216 this would be $2\text{ }\times \text{ }108$ then again we will break 108 into smaller factors, for 108 this would be $2\times 54$ and so on. We will continue this process until we get all the factors as a prime number.
Let’s do it step by step to understand it more clearly :-
$216\text{ }=\text{ }2\text{ }\times \text{ }108$
$\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }54$
$\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }27$
$\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\times \text{ }9$
$\text{ }=\text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }2\text{ }\times \text{ }3\times \text{ }3\text{ }\times \text{ }3$
Now, we will express 216 as a product of power of prime factors.
So, the final answer would be $\text{ }{{2}^{3}}\text{ }\times \text{ }{{3}^{3}}$.

Note: 216 is a composite number where composite number is a positive integer that has at least one factor other than 1 and number itself. Here in this example we can clearly see that 216 is a composite number as it has factors other than 1 and 216.
Also, $216$ is a perfect cube. As $216$ can be expressed as $\text{ }{{2}^{3}}\text{ }\times \text{ }{{3}^{3}}$ . Since the power of 2 and 3 are same so we can write it as $\text{ }{{\left( 2\times 3 \right)}^{3}}={{6}^{3}}$
Therefore 216 is a perfect cube.
So we can also use the prime factorization method to check whether the number is a perfect cube or perfect square.