Estimating Square Root and Cube Root of Numbers

In case we have a number with many digits, it will be very difficult for us to calculate the square roots and cube roots of the number by prime factorization and long division methods. Hence we try to estimate the value of cube roots and square roots for these numbers.

In prime factorization, we find the factors of a number. Numbers such as 25, 700, 368, etc are easy to factorize. In case the number of digits increase the factorization becomes difficult. In such cases estimating square root and cube root is a good option.

For estimating the number of digits in the square root of a number, we use method of bars. For example:

\( \sqrt{961} \) = 31 and \( \sqrt {1369} \) = 37

The bar is placed on pair of digits starting from the rightmost digit.

\( \overline{9} \;\overline{61}\; and\; \overline{13}\; \overline{69} \)

In both the numbers above, we have 2 bars therefore their square root will have two digits.

In order to estimate the square root of a number consider the following example:

Number whose square root is to be determined = 247

Since: 100 < 247 < 400, and \( \sqrt[2]{100} \) = 10 and \( \sqrt[2]{400} \) = 20.

Taking the square root we get: 10 < \( \sqrt[2]{247} \) < 20

We are not very close to the number yet.

We also know that \( 15^2 \) = 225 and \( 15^2 \) = 256, therefore 15 < \( \sqrt[2]{247} \) < 16

The number 247 is much closer to 256 than 225.

Hence \( \sqrt[2]{247} \) is approximately equal to 16.

Estimating cube root of a cube number

If a large number is given and stated that it is a perfect cube then we can use the following method to calculate its cube root by the method of estimation.

1. Take any cube number such as 29791 whose cube root is to be determined and start making a group of three digits starting from the rightmost digit.

2. The first group will give us the unit’s digit of the required cube root. The number 791 ends in 1 therefore the last digit will be 1 because unit’s digit is 1 only when the cube root also ends with 1.

3. Now taking the second group, it is between \(3^3\) = 27 and \(4^3\) = 64. We take the smaller value therefore the digit at tenth place is 3.

The required cube root = \( \sqrt[3]{29791} \) = 31.