Question

How do you find the square root of $343$ ?

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 $343$ is not a perfect square. Now, to simplify the square root of $343$, we first do the prime factorization of the number and take the factors occurring in pairs outside of the square root radical.

Complete answer: Given, $343$
$343$ can be factored as,
$343 = 7 \times 7 \times 7$
Now, expressing the prime factorization in powers and exponents, we get,
$343 = {7^3}$
We can see that $7$ is multiplied three times and hence the power of $7$ is three.
Now, $\sqrt {343} = \sqrt {{7^3}} $
We know that ${7^3} = {7^2} \times 7$. So, ${7^3}$ can be written as ${7^2} \times 7$.
Now, $\sqrt {343} = \sqrt {{7^2} \times 7} $
Since we know that ${7^2}$ is a perfect square. So, we can take this outside of the square root we have,
So, $\sqrt {343} = 7 \times \sqrt 7 $
Since $7$ is not a perfect square, we can multiply this and keep it inside the square root,
$ \Rightarrow \sqrt {343} = 7\sqrt 7 $
This is the simplified form of $\sqrt {343} $

Note:
Square root of a number can be found using a number of methods like long division method and prime factorization method. We must adopt the above given technique to find the square root of any given number as it is the simplest of all. Pairs of factors can be taken out of the square root radical in order to simplify the given square root expression. Here $\sqrt {} $ is the radical symbol used to represent the root of numbers. The number under the radical symbol is called radicand.