We mentioned before that when you square a number, you’ll get a square number or a perfect square. You can use the perfect square to find out what numbers were multiplied together to get the perfect square. The process of using the perfect square to get back to those numbers is finding the square root of a number. For example, 3 * 3 = 3^2 = 9. 9 Here is perfect square. You can take the square root of 9 to get back to 3. You can see that squaring a number and taking the square root of the result are actually inverse processes.
There are different ways to do just that.
Finding Square Root by Inference
One way of finding the square root of numbers is by inference. For example, you know that 3 * 3 = 3^2 = 9 so you can infer that sqrt(9) = 3. This method works for small numbers that you are already familiar with. But what about sqrt(106929)? You need to have a way to calculate that.
Every number has two square roots, a positive and a negative one. For example. 3^2 = 9 and also (-3)^2 = 9. So then you can say that sqrt(9) = +/- 3. In this discussion, we’ll restrict ourselves to the positive square root or the principal square root of numbers. We’ll talk about the negative square root elsewhere.
Finding Square Root by Repeated Subtraction
Earlier we showed that every perfect square can be expressed as the sum of some consecutive odd natural numbers starting from 1. For example, 9 can be expressed as (1 + 3 + 5). Since you had to add the first three odd consecutive natural numbers starting from 1 together to get 9, you can write 9 as 3^2. You can use this property of perfect squares to find out the square root of every perfect square. For example, in the case of the same perfect square 9, to find out how many odd consecutive natural numbers starting from 1 have to be added together to get 9, all you need to do is subtract those numbers starting from 1 repeatedly from 9 until you get to 0. The number of those numbers will give you the value of n in n^2 and then you can say that for example 9 is the same thing as 3^2. In the case of 9, 9 – 1 – 3 -5 = 0. So I had to subtract the first three odd consecutive natural numbers starting from 1 to get to 0, meaning that 9 is the same thing as (1 + 3 + 5). Since the number of those numbers was 3, then I express 9 as 3^2.
The method of repeated subtraction works well in the case of relatively small numbers probably all the way to 100 or so. If you get above 100, the number of subtractions required would take much time. So you’ll need another method to find the square root of larger numbers like 106929.
Finding Square Root by Prime Factorization
Using prime factorization works because every number is a collection of certain prime factors multiplied together. For example, 9 is two factors of 3. 18 is two factors of 2 and one factor of 3. In the same way, you can express any number in terms of some prime factors.
Now if you pay attention to the prime factorization of any number and its perfect square, you’ll see an interesting pattern. For example, 9 will be factorized as two factors of 3 multiplied together. On the other hand, 81 will be factorized as 4 factors of 3 multiplied together and 81 = 9^2. 81 has twice the number of factors of 3 that 9 has. In the same way, 6 has one factor of 2 and one factor of 3. 36 has two factors of 2 and two factors of 3. Meaning that 36 has twice the factors of 2 and 3 that 6 has and 36 = 6^2. That means that the square of any number will have twice all the factors that that number has. You can use this pattern to find the square root of any perfect square by factorizing it. Simply pick any perfect square, factorize it and take half the number of any factors that it has and multiply them together and you’ll get the square root of that number.
As an example, 36 = 2 * 2 * 3 * 3. And so sqrt(36) = 2 * 3. 36 had two factors of 2, sqrt(36) has one factor of 2. 36 had two factors of 3. sqrt(36) had one factor of 3. Multiply them together: 2*3 = sqrt(36) = 6.