Question

How do you simplify the square root of 196?

Answer 1

Hint: We are given $\sqrt{196}$. We have to simplify. We will learn how to simplify such an expression, we will work with some examples to get the understanding, then we work on our problem, we start with a prime factorisation of 196 and then we take a pair out of the root leaving all un paired inside it and then we simplify and solve.

Complete step by step solution:
We are given an equation, a term with square root, inside it we have 196. To learn how to simplify such a question, we will learn some examples related to this.
To solve such a problem where we have radical $\left( \sqrt{{}} \right)$ in the numerator only and denominator is simply one, this type of question is mainly based on prime factorization to simplify.
Like we consider we have $\sqrt{8}$ .
So, we have 8 inside square root, we factor 8 into its prime factor so we get –
$8=2\times 2\times 2$
$\Rightarrow \sqrt{8}=\sqrt{2\times 2\times 2}$
Now properties of square root is that only a pair of terms can be taken out of it as we can, we have 3 pieces of 2, so 2 pieces of 2 will make one pair so it will come out and the remaining one will stay inside the root. So,
$\begin{align}
  & \sqrt{8}=\sqrt{2\times 2\times 2} \\
 & =2\times \sqrt{2} \\
\end{align}$
So, we get –
$\sqrt{8}=2\sqrt{2}$
Similarly say if we have $\sqrt{16}$ .
So,
$\Rightarrow \sqrt{16}=\sqrt{2\times 2\times 2\times 2}$
Here we get fully paired up so our term will come out and there will be nothing to root.
$\begin{align}
  & \sqrt{16}=2\times 2 \\
 & =4 \\
\end{align}$
So, the simplification of $\sqrt{16}$ is ‘4’.
Now we work on our problem, we have $\sqrt{196}$ .
So, we start our solution by finding the prime factor of 196.
So, we get –
$\Rightarrow 196=2\times 2\times 7\times 7$ .
We can see that in the prime factorization of 196, we have one pair of ‘2’ and one pair of ‘7’, so they will come out of the square root.
That is we have –
$\Rightarrow \sqrt{196}=\sqrt{2\times 2\times 7\times 7}$
One ‘2’ and one ‘7’ will come out and we leave with nothing inside the root.
So, $\sqrt{196}=2\times 7$
By simplifying, we get –
$=14$
So, the square root of 196 is 14.

Note: Remember that to find the square root or to simplify the term inside the square root we start by factorization and then making the pair of two.
If we have cube root $\left( \sqrt[3]{{}} \right)$ then we make a pair of three terms after the factorization.
And similarly for different (nth root) we make different pairs before taking out, also remember we factor it carefully.