How Do You Compute a Square Root?

Determining the square root of a number can be achieved through various methods. In this article, we will explore three popular techniques: Heron's (or Newton-Raphson) method, the Long Division method, and a trick for perfect radicands. These methods serve as valuable tools for mathematicians, scientists, and anyone needing to perform square root calculations.

Heron's (or Newton-Raphson) Method

One of the most well-known iterative methods for computing square roots is Heron's method, also known as the Newton-Raphson method. This method involves an iterative process that refines a guess towards the actual square root with each step. The formula for this method is:

xn 1 0.5(x_n a / x_n)

where a is the radicand (the number whose square root is sought) and x_n is the closest perfect square to the radicand. Here's a step-by-step example:

Example: Calculate the square root of 974169.1. Start with an initial guess: x_1  1000, since 10002  1,000,000 is close to 974169.2. Iteration 1: x2  0.5(1000   974169 / 1000)  0.5(1000   974.169)  987.08453. Iteration 2: x3  0.5(987   974169 / 987)  0.5(987   987.0845)  987

Since the value converges quickly, we can conclude that the square root of 974169 is 987.

The Long Division Method

A visual method for calculating square roots is the long division method. This approach works by breaking the radicand into digit pairs and performing a division process to find the square root step by step. The formula for each step is based on the following expression:

20 ab b2

Let's go through an example with 117.89.

Example: Calculate the square root of 117.89.1. Split the radicand into digit pairs: 11 middot; 78 middot; 9.2. Find the largest perfect square that fits into the first digit or group: 1 fits into 1, with no remainder. The next two digits 17 are our first dividend: 17 dagger; 20 dagger; 1  0.85.3. Continue the process for the first decimal place: 1789 dagger; 20 dagger; 10  8.945. The second digit after 1 could be 8, as 20 middot; 10 middot; 882  1664, which fits into 1789. The remainder is 125, and the next dividend is 12500 dagger; 20 dagger; 108 ≈ 5.787. This indicates that the next digit could be 5.4. Using the formula: 20 middot; 108 middot; 552  10825, which fits. Therefore, the square root of 117.89 starts with 10.85, and 10.852 is 117.7225, which is very close to our radicand.

A Trick for Perfect Radicands

For radicands that are perfect squares and have no more than 4 to 6 digits, there is a neat trick you can use. You only need to know the first few perfect squares (up to 202 or 322) to apply this method.

Example: Calculate the square root of 26896. Split the radicand into digit pairs: 26 middot; 89 middot; 6. The largest perfect square that fits into the first digit or pair is 162256, so the first two digits of the square root are 16. Calculate 16 × 16 256, which is less than 268. Therefore, the last digit of the square root must be 4 (the smaller option). Thus, the square root of 26896 is 164.

Key Takeaways

These methods provide practical and efficient ways to compute square roots. Whether you're using an iterative approach, a long division process, or a quick trick for perfect radicands, these techniques can help you determine square roots with accuracy and speed.