Question

Find the square root of a given number by the prime factorization method: 784.

Answer 1

Hint: First, we will understand the concept of prime factorization method i.e. finding which prime numbers multiply together to make the original number. Then we will find the first prime factors of 784 and group it into pairs of two same digits. For example: $ 4=\underline{2\times 2} $, so, the square root of 4 will be 2. Similarly, we will be getting a square root of 784.

Complete step-by-step answer:
Here, we will first understand what is meant by prime factorization.
Prime factorization is finding which prime numbers multiply together to make the original number. To understand this, we will take an example: Suppose we have to find factors of number 12. So, first we will see whether the number is divisible by 2 or not. So, we will get $ 12=2\times 6 $ . Now, we will take 6 and see whether it is divided by 2 or not. So, we will get $ 12=2\times 2\times 3 $ . Now, we know that 3 is a prime number so no need to further solve this. Hence, we get our prime factors 2 and 3.
So, here also, we will first divide 784 using prime factorization. We will get as
 $ \begin{align}
  & 2\left| \!{\underline {\,
  784 \,}} \right. \\
 & 2\left| \!{\underline {\,
  392 \,}} \right. \\
 & 2\left| \!{\underline {\,
  196 \,}} \right. \\
 & 2\left| \!{\underline {\,
  98 \,}} \right. \\
 & 7\left| \!{\underline {\,
  49 \,}} \right. \\
 & 7\left| \!{\underline {\,
  7 \,}} \right. \\
\end{align} $
So, we can write it as $ 784=2\times 2\times 2\times 2\times 7\times 7 $
Now, we have to find the square root of 784. So, we will group the same numbers in pairs of two i.e. \[784=\underline{2\times 2}\times \underline{2\times 2}\times \underline{7\times 7}\] . So, taking one digit from three pairs, we will get as \[2\times 2\times 7=28\] .
Thus, the square root of 784 is 28.

Note: Always remember that in this type of problem we must do verification to check whether our answer is correct or not. So, here if we do square of 28, then we will be getting 784 which is the correct answer. Also, we have taken prime number as 2 and then solved it but if we direct 7 first and then try to solve, at last we will get the same answer i.e. 28.