Question

How do you find the square root of $320$ ?

Answer 1

Hint:Square root of a number is a value, which when multiplied by itself gives the original number. Suppose, ‘x’ is the square root of ‘y’, then it is represented as $x = \sqrt y $ or we can express the same equation as ${x^2} = y$ . Here we can see that $320$ is not a perfect square. Now, to simplify the square root of $320$, we first do the prime factorization of the number and take the factors occurring in pairs outside of the square root radical.

Complete step by step answer:
Given, $320$ can be factorized as,
$320 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5$
Now, expressing the prime factorization in powers and exponents, we get,
$320 = {2^5} \times 5$
We can see that $2$ is multiplied five times and hence the power of $2$ is five.
Now, $\sqrt {320} = \sqrt {{2^5} \times 5} $
We know that ${2^5} = {2^4} \times 2$. So, ${2^5}$ can be written as ${2^4} \times 2$.
Now, $\sqrt {320} = \sqrt {{2^4} \times 2 \times 5} $
Since we know that ${2^4}$ is a perfect square. So, we can take this outside of the square root we have,
So, $\sqrt {320} = {2^2} \times \sqrt {2 \times 5} $
Since $2$ and $5$ are not perfect squares, we can multiply this and keep it inside the square root,
$ \therefore \sqrt {320} = 4\sqrt {10} $
This is the simplified form.

Note: Here $\sqrt {} $ is the radical symbol used to represent the root of numbers.The number under the radical symbol is called radicand. The positive number, when multiplied by itself, represents the square of the number. The square root of the square of a positive number gives the original number. To find the factors, find the smallest prime number that divides the given number and divide it by that number, and then again find the smallest prime number that divides the number obtained and so on. The set of prime numbers obtained that are multiplied to each other to form the bigger number are called the factors.