Let’s take a look at how we would find the square root of a perfect square. You might ask the question, what is a perfect square? Before we answer that, lets back up and answer, what is the square of any number? The square of a number is the product of a number and itself. A perfect square is a product of an integer and itself. Let’s look at some examples of a perfect square. Here are a few examples.
Twenty-five is the product of five times five. Twenty-five equals five squared. Four is also a perfect square. Four is the product of two times two, also known as two squared. Can you think of an integer multiplied by itself that equals forty-nine? It’s seven, seven times seven equals forty-nine. Forty-nine equals seven squared. And now for one hundred, do you have any ideas? Ten times ten equals one hundred, or ten squared.
If you remember the title of the video though, square roots of perfect squares, we’re not just talking about perfect squares. We wanna talk about how to find the square roots of perfect squares. Let’s start by defining square roots. The factors multiplied to form squares are called square roots. Let me read that one more time. The factors multiplied to form squares are called square roots. Let’s go back to our example of twenty-five. Twenty-five equals five squared, or five times five. The factors multiplied to form the square of twenty-five is a five. So we say, the square root of twenty-five is five.
We use this symbol to denote finding the square root of something. This symbol is called a radical sign. These are the symbols we would use if we wanted to say the square root of twenty-five is five. First, we have the radical sign, put the perfect square inside the radical, and our solution, five, is the square root of twenty-five. Here are two examples.
The first one says, find the square root of eighty-one. And the second one says, find the square root of two hundred and twenty-five.
Let’s start here. We know we’re looking for some integer multiplied by itself. I know ten multiplied by itself equals one hundred and that eighty-one is smaller than that. Then I recognize nine times nine equals eighty-one; the square root of eighty-one has to be nine. So we have the final answer. The square root of eighty-one equals nine. Let’s try our next example.
You’re thinking the square root of two hundred and twenty-five is some number multiplied by itself that equals two hundred and twenty-five. And twelve times twelve equals one hundred and forty-four, thirteen times thirteen equals one hundred and sixty-nine. You probably memorized those values at some point. But now you’re just wondering what can I do, do I have to keep guessing and checking my answer? There are some strategies that can help you solve this mentally.
Notice that in twelve times twelve equals one hundred and forty-four, you see that two times two equals four and that’s the last digit in the number. You can also notice in thirteen times thirteen equals one hundred and sixty-nine, the same pattern is there. So we’re gonna be looking for something that multiplies together and has a five in that position. If you look at one times one, two times two, three times three, four times four, all the way to five times five, the only thing that ends with an integer of five is five times five.
We’re looking for a number that’s larger than thirteen and ends in five. So it would be smart to check fifteen as your next value. In fact, fifteen times fifteen does equal two hundred and twenty-five, which makes the square root of two hundred and twenty-five, fifteen. Now, you’re ready to use mental math strategies to recognize and find the square roots of perfect squares.