Principal Square Root: Definition & Example

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• 0:00 What Is Squaring?
• 1:28 What Are Square Roots?
• 2:15 What Are Perfect Squares?
• 3:01 Principal Square Roots
• 4:25 Lesson Summary
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Lesson Transcript
Instructor: Lance Cain
In this lesson, we will refresh our understanding of squaring a number and then explore the inverse operation called square root and define the principal root.

What Is Squaring?

First, remember that squaring a number means that you multiply the number times itself. For example, 2 squared equals 2 x 2, which equals 4.

But, why is it called squaring a number? Good question. Take a look at the image below. If you draw small squares and line them up according to the number you are multiplying, each group of small squares makes one large square. In this way, 2 squared (or 2 x 2) can be shown by putting two small squares across the top and two small squares down the side. You need four total squares to complete the 2 x 2 larger square. This is a visual representation of 2 x 2 = 4. Similarly, 3 x 3 can be illustrated with three small squares across the top row and three small squares in the first column, and then fill in the rest of the diagram to form the larger square; it takes nine total small squares. Thus, 3 x 3 = 9.

4 x 4 takes 16 small squares, 5 x 5 takes 25 squares, and so on. If we had room here, we could arrange 10 small squares in a row and 10 in the first column, and then fill in the larger square. How many small squares would be needed for 10 x 10? That's right, 100 small squares. 10 squared = 100. Think of carpeting the floor of a room that is 10 feet wide and 10 feet long, you would need 10 x 10 = 100 square feet of carpet.

What Are Square Roots?

Now, let's reverse the squaring process. For example, what if I told you I had a large square made up of 100 small squares, could you tell me how many rows and columns there would be? Right again, 10 rows and 10 columns, 10 x 10 = 100. That is how finding the square root works, finding a number times itself that gives you the result you are looking for.

Above is the square root symbol. This is read as the square root of a. Conversely, y is the square root of a if y squared = a. In other words, y x y = a. For example, what is the square root of 49? Answer, sqrt (49) = 7 because 7 x 7 equals 49.

What Are Perfect Squares?

As demonstrated in the table below, each number has a positive and a negative root. For simplicity, only perfect squares are shown, that is only numbers that have whole number roots. For example, 4 x 4 = 16, so the square root of 16 is 4 (a whole number); therefore, 16 is a perfect square. But every positive number has a square root, it's just that the square roots of most numbers are not whole numbers. Take the number 6 for example. Using a calculator, you'll find that the square root of 6 is about 2.45, which is not a whole number; therefore, 6 is not a perfect square. The square root of 8 is approximately 2.83, not a whole number; therefore, 8 is not a perfect square.

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