Perfect Squares and Square Roots

Hello, in this video, we will explore how to simplify square roots and find perfect squares.

Terminology

The symbol in math that we use to represent square roots is called a radical, and it looks like this: \(\sqrt{}\). The number that appears under the radical is called the radicand.

For example, in this expression, \(\sqrt{20}\), which we read as, “the square root of twenty,” 20 is the radicand. Just like the fact that 20 is the same as \(\frac{20}{1}\), or \(20^{1}\), but we don’t write the 1, because it is a given, in square roots there is something similar.

There is a given 2 in the bent arm of the radical that we do not normally write when we are looking for a square root, \(\sqrt[2]{}\). This number is called the index and it determines what root of the radicand we are trying to find. For example, this: \(\sqrt{20}\), is asking for the square root, or the second root of 20 and this: \(\sqrt[3]{125}\), which we read as “the cube root of one hundred twenty-five,” is asking us to find the third root of 125.

I want you to practice this on your own. Pause the video and label the different parts of this expression: \(\sqrt[4]{200}\).

Think you’ve got it? The root symbol is the radical. The index is 4 because it’s the number in the bent arm of the radical. And the radicand is the number under the radical symbol, so in this case, 200.

 
Great work!

Finding the Square Root

Now let’s talk about how we find the square root of a number. To do this, we ask ourselves, “what number multiplied by itself will give us the radicand?”

For example, what is the square root of 25: \(\sqrt{25}\)? The factors of 25 are \(5\times 5\), which means that the square root of 25 is 5. This also happens to be what we call a perfect square. A perfect square is when we are simplifying a square root and nothing remains under the radical.

What are the square roots of 36, 81, and 144? Pause the video and try these on your own. When you’re finished, we’ll look over them together.

 
We get these nice, pretty numbers because we are taking the square roots of perfect squares. But what if we aren’t given a perfect square? Well, then we will have to simplify the square root.

Simplifying Square Roots

When simplifying a square root, we will get all the perfect squares out from under the radical and whatever is remaining from the factors of the radicand stays under the radical. Let’s look at an example.

Simplify the square root of 40: \(\sqrt{40}\).

We will start by finding the factors of 40, which are \({2}\times {2}\times {2}\times{5}\).

Since \(2\times{2}\) creates a perfect square, we can bring the 2 to the front and what remains under the radical is \(2\times{5}\).

There are no more perfect squares to take out, so we simply multiply these numbers together to get our new radicand.

Therefore, the square root of 40 is equal to 2 square root of 10: \(\sqrt{40}\) = \(2\sqrt{10}\).

Now I want you to try one. Simplify the square root of 96: 96. Pause the video here and simplify. When you’re done, we’ll take a look at it together.

Think you’ve got it? First, we need to find the factors of 96.

 
It is easy to see the perfect squares when the factors are written in this form. We can pull out two sets of 2 from our radicand, which will leave us with \(2\times{2}=4\) in front of our new radical. We still have a 2 and a 3 from our factor list, so we multiply these numbers together to get 6, and this is our new radicand. Therefore, the most simplified form is:

 
I hope this video on square roots and perfect squares was helpful. Thanks for watching, and happy studying!

Frequently Asked Questions

Perfect Squares and Square Root Practice Questions