How to Calculate Square and Cube Roots Without a Calculator

Calculating roots like square roots √ or cube roots can be approached using several methods. Here are some techniques for both square roots and cube roots to help you approximate these values without a calculator.

Square Roots √

Have you ever needed to calculate the square root of a number, say √3, without a calculator? It is possible, and here are a few methods that can guide you through the process.

Estimation

Identify two perfect squares between which the number lies. For example, for √3: 1^2 1 2^2 4 Therefore, 1 √3 2.

Estimate a value. Since 3 is closer to 4, you might guess around 1.7.

Babylonian Method (Heron's Method)

Start with an initial guess x_0. For √3, you might start with x_0 1.5.

x_{n1} (x_n (3 / x_n)) / 2

Repeat until you reach a satisfactory level of accuracy:

x_1 (1.5 (3 / 1.5)) / 2 (1.5 2) / 2 1.75 x_2 (1.75 (3 / 1.75)) / 2 ≈ 1.732

Cube Roots

Let's look at a similar process for cube roots. For example, calculating the cube root of 4:

Estimation

Identify two perfect cubes. For 4:

1^3 1 2^3 8 Therefore, 1 ?4 2.

Estimate a value. Since 4 is closer to 1^3, you might guess around 1.5.

Refine your guess:

1.5^3 3.375 1.6^3 4.096

So, the cube root of 4 is between 1.5 and 1.6.

Using a Method Similar to the Babylonian Method

Start with an initial guess, say x_0 1.5 for the cube root of 4.

x_{n1} (2x_n - (4 / x_n^2)) / 3

Repeat until you reach a satisfactory level of accuracy:

x_1 (2 * 1.5 - (4 / 1.5^2)) / 3 ≈ 1.587

Conclusion

These methods allow you to approximate square and cube roots fairly accurately without a calculator. With practice, you can refine your estimates more quickly and accurately.

Additional Methods for Procedural Calculation

Another method involves long division with the binomial formula, which is specific for square roots and cube roots.

Example: Cube Root of 4

In the case of the cube root of 4, as a0 in the first step, we need to find the largest perfect cube fitting into 4, which is 1^3 1.

The remainder is 3, which when multiplied by 1000 gives the first dividend. Using the binomial formula, we calculate:

3000 / (300 / 1^2) 10. This suggests b could be 10 but is constrained to be 9 or less. Testing further, we find:

300 * 15^2 * 9 30 * 15 * 9^2 9^3 644679 300 * 15^2 * 8 30 * 15 * 8^2 8^3 614248 300 * 15^2 * 7 30 * 15 * 7^2 7^3 583717 300 * 15^2 * 6 30 * 15 * 6^2 6^3 553186 300 * 15^2 * 5 30 * 15 * 5^2 5^3 522655

Thus, b 5, and the remainder is 625 * 1000 625000 for the next dividend.

Further iterations can be carried out to refine the approximation.