Question

Find the value of $\sqrt7$ up to six decimal places.

Answer 1

Hint: To determine the square root of a number we can either use the long division method or the prime factorisation method. As here in the question we are asked to find the root of 7 to six decimal places, long division method would be a wise choice.

Complete step-by-step answer:
For numbers like $7$ where it is difficult to find the factors, we use the division method for finding the root. First, we make a pair of digits starting from the unit place. Then we need to find the largest perfect square smaller than the number formed by the pair of digits appearing first from the right.
$2\overset{2}{\overline{\left){\begin{align}
  & 7 \\
 & \underline{\text{4 }} \\
 & 3 \\
\end{align}}\right.}}$

Now double the quotient you got in the first step and make it the tenth place of a number, now find a unit place to the number such that the number into its unit place is equal to or closest to the number you obtained as the remainder of the previous steps. Also, make the unit digit of the number, the unit digit of the quotient as well. Repeat the steps till you get the remainder zero. However, if you introduce a decimal point to the quotient, unlike normal division instead of one zero, pair of zeroes are introduced at the end of the remainder of the previous step.

Therefore, the root of 7 is 2.645751, correct up to six decimal places.

Note: While calculating square roots and cube roots, prime factorization is the easiest method. But it takes time and difficult to use for the questions where the square roots are not integers. Hence, other methods should also be learnt, so that they can be used while solving problems in cases where time plays an important role.