Question

What is the square root of 82?

Answer 1

Hint: First we will check if we can use the prime factorization method to find the square root or not. If we will be able to write all the prime factors as exponent equal to 2 then we will be able to find the root easily. If this method fails then we will apply the long division method to get our answer in decimal up to three places.

Complete step by step answer:
Here we have been asked to find the square root of 82.
First let us check if we can use the prime factorization method to get the root. So writing 82 as the product of its primes we get,
$\Rightarrow 82=2\times 41$
As we can see that we will not be able to write the factors with their exponent equal to 2 it means 82 is not a perfect square. Now, we need to apply the long division process to find the square root. So, let us see each step while finding the root.
Step (1): Starting from the rightmost digit we will form pairs of two digits till all the digits are paired or a single digit is left at the leftmost place, here we have only two digits in 82 so they will get paired.
Step (2): We will select a number whose square will be less than or equal to 82. Here we have to select 9. Now, we will write 9 as the divisor and the quotient and subtract the product from 82. The remainder will be 1.

Step (3): Now, the new divisor will be the sum 9 + 9 = 18. Clearly we can see that we cannot divide 1 by 18 so we will take a decimal point after 9 is the quotient and put two 0’s after 1 in the remainder.

Step (4): Now, we need to select a digit after 18 such that when we will take the product of this selected digit with the new divisor we will get a number less than 100. Clearly we can see that there will not be any such positive integer so we will take 0. Now the new divisor will become 180 + 0 = 180 and the new dividend will become 10000 formed by again putting two 0’s after 100. The quotient will become 9.0.

Step (5): In this step we will take 5 after 180 such that the new divisor becomes 1805 and we will multiply 1805 with 5 and subtract this product from 10000 to get the new remainder 975. The new quotient will be 9.05 and the new divisor will be 1805 + 5 = 1810.

Step (6): Again we will perform a similar process like step 4 and add two 0’s after 975 and take 5 after 1810 to get the new divisor as 18105 and the new quotient as 9.055 which will be our answer.

Hence, the square root of 82 is 9.055 correct up to three places of decimal.

Note: You must remember the long division method to find the square root because this method is very useful in determining the square roots of numbers which are not perfect squares. Note that here we had only two digits in 82 so they got paired easily, if we have a number like 457 then we will form a pair of 7 and 5 while leaving the digit 4 alone and follow the same process as above.