9.2 Simplify square roots

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that $\sqrt{50}$ is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use [link] .

But what if we want to estimate $\sqrt{500}$ ? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.

A square root is considered simplified if its radicand contains no perfect square factors.

So $\sqrt{31}$ is simplified. But $\sqrt{32}$ is not simplified, because 16 is a perfect square factor of 32.

Use the product property to simplify square roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that ${(ab)}^m=a^m{b}^m$ . The corresponding property of square roots says that $\sqrt{ab}=\sqrt{a}·\sqrt{b}$ .

We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in [link] .

How to use the product property to simplify a square root

Simplify: $\sqrt{50}$ .

Solution

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Notice in the previous example that the simplified form of $\sqrt{50}$ is $5\sqrt{2}$ , which is the product of an integer and a square root. We always write the integer in front of the square root.

We could use the simplified form $10\sqrt{5}$ to estimate $\sqrt{500}$ . We know 5 is between 2 and 3, and $\sqrt{500}$ is $10\sqrt{5}$ . So $\sqrt{500}$ is between 20 and 30.

The next example is much like the previous examples, but with variables.

We follow the same procedure when there is a coefficient in the radical, too.

In the next example both the constant and the variable have perfect square factors.