Question

Find the square root of 784 by prime factorisation method.

Answer 1

Hint: Resolve the number 784 into its prime factors. Group the factors which appear twice in the form of a square quantity. Then apply square root on those groups to obtain these prime factors and multiply them to find the square root of the number 784.

Complete step-by-step answer:
Prime Factorisation method is a method to obtain the square root of a number, where the number itself is resolved into groups of two to form a square quantity and then applying square root on each of these groups to obtain the prime factors of the square root of that number. The product of these factors give the square root of the given number.
Suppose, 784 is the given number.
Now,
$\begin{array}{rl}& 2|\underset{―}{784}\\ & 2|\underset{―}{392}\\ & 2|\underset{―}{196}\\ & 2|\underset{―}{98}\\ & 7|\underset{―}{49}\\ & 7|\underset{―}{7}\\ & \text{ 1}\end{array}$
 Thus, 784 can be factorized as follows,
784 = 2 x 2 x 2 x 2 x 7 x 7
       $={\text{2}}^2\text{ x }{\text{2}}^2\text{ x }{\text{7}}^2$
Applying square root on both sides, we get,
$\begin{array}{rl}& \sqrt{784}\text{ = }\sqrt{2^2\text{ x }{\text{2}}^2\text{ x }{\text{7}}^2}\\ & \text{ = 2 x 2 x 7}\\ & \text{ = 28}\end{array}$
Thus, by prime factorisation method, we obtain the square root of 784 to be 28.
The correct answer is 28.

Note: In the factors of 784, there are two groups of the prime factor 2, i.e. $2^2\text{ x }{\text{2}}^2$. It can be simply expressed as $4^2$, as then also it will yield the same result. But 4 is not a prime factor, so 784 is not expressed in terms of $4^2$. Instead we use $2^2\text{ x }{\text{2}}^2$ because 2 is a prime factor.