Question

How do you find the square root of 1414?

Answer 1

Hint: Use long division method.This problem is step by step. First divide the number 1414 in pairs from right. Find the largest perfect square less than the right most pair. Subtract the below number from the pair and write the difference below it. Multiply the number on top by 2 and use that number and the bottom most number to write a problem. Do these steps 1 more time to get an answer.

Complete step by step solution:
The given question is to find the square root of 1414.
We will solve this problem using the long division method which is specifically used to solve problems like this.
Please note that, we will use this method till one decimal accuracy only. So let’s start and find the square root of 1414.
Step 1:
Divide the number 1414 in pairs. Group two numbers at a time starting from the right hand side and add two zeros at the end of the number.
$1414 \to \left[ {14} \right]\left[ {14} \right]\left[ {00} \right]$
Step 2:
Now, start from the right most pair and try to find the largest perfect square less than that number
Here, the right most pair is 14 and we know that the largest perfect square less than 14 is 9. And the square root of 9 is 3
So, write 3 above and 9 below the pair 14
$
   \to \left[ 3 \right] \\
  1414 \to \left[ {14} \right]\left[ {14} \right]\left[ {00} \right] \\
   \to \left[ 9 \right] \\
$
Step 3:
Now, subtract the below number from the pair and write the difference below it.
After that, write the next pair, that is the second pair after that difference.
$
   \to \left[ 3 \right] \\
  1414 \to \left[ {14} \right]\left[ {14} \right]\left[ {00} \right] \\
   \to \left[ 9 \right] \\
   - \_\_\_\_ \\
   \to \left[ 5 \right]\left[ {14} \right] \\
$
Step 4:
Now, multiply the number on top by 2 and use that number and the bottom most number to write a problem like this:-
$6? \times ? \leqslant 514$
Now, we know that the maximum number that can take place of that question mark is 7
So, we will write the number 7 after 3 on the top most row and write the product of $67 \times 7 = 469$ below 514.
$
   \to \left[ 3 \right]\left[ 7 \right] \\
  1414 \to \left[ {14} \right]\left[ {14} \right]\left[ {00} \right] \\
   \to \left[ 9 \right] \\
   - \_\_\_\_ \\
   \to \left[ 5 \right]\left[ {14} \right] \\
   \to \left[ 4 \right]\left[ {69} \right] \\
$
Step 5:
Subtract 469 from 514 and write the difference below it and again write the next of the original pair behind it.
\[
   \to \left[ 3 \right]\left[ 7 \right] \\
  1414 \to \left[ {14} \right]\left[ {14} \right]\left[ {00} \right] \\
   \to \left[ 9 \right] \\
   - \_\_\_\_ \\
   \to \left[ 5 \right]\left[ {14} \right] \\
   \to \left[ 4 \right]\left[ {69} \right] \\
   - \_\_\_\_\_\_\_\_ \\
   \to \left[ {00} \right]\left[ {45} \right]\left[ {00} \right] \\
\]
Step 6:
Again multiply the number on top by 2 and use that and the bottom most number to write problem like this
$74? \times ? \times 4500$
Again, we can say that the maximum number we can write here is 6
Therefore, write 6 on top.
$
   \to \left[ 3 \right]\left[ 7 \right]\left[ 6 \right] \\
  1414 \to \left[ {14} \right]\left[ {14} \right]\left[ {00} \right] \\
   \to \left[ 9 \right] \\
   - \_\_\_\_ \\
   \to \left[ 5 \right]\left[ {14} \right] \\
   \to \left[ 4 \right]\left[ {69} \right] \\
   - \_\_\_\_\_\_\_\_ \\
   \to \left[ {00} \right]\left[ {45} \right]\left[ {00} \right] \\
$
That's it since we decided to find accuracy till 1 decimal point. Place 1 decimal left from the right. That will be our answer
Hence, $\sqrt {1414} = 37.6$

Note:
Note, this step should be used when there is no integral square root of the number. Those numbers are also known as non-perfect squares. Numbers like $2,4,9,16,25,36,49$ etc none of are perfect squares. Whereas $2,4,9,16,25,36,49$ etc are perfect squares.