5.5 Simplify Square Roots
Learning Objectives
By the end of this section, you will be able to:
- Use the Product Property to simplify square roots
- Use the Quotient Property to simplify square roots
In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use in (Simplify and Use Square Roots).
But what if we want to estimate ? 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.
Simplified Square Root
is considered simplified if has no perfect square factors.
So is simplified. But 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 . The corresponding property of square roots says that .
Product Property of Square Roots
If a, b are non-negative real numbers, then .
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 (Example 1).
EXAMPLE 1
Simplify: .
| Step 1. Find the largest perfect square factor of the radicand. | is the largest perfect square factor of | |
| Rewrite the radicand as a product using the perfect square factor. | ||
| Step 2. Use the product rule to rewrite the radical as the product of two radicals. | ||
| Step 3. Simplify the square root of the perfect square. |
TRY IT 1.1
Simplify: .
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TRY IT 1.2
Simplify: .
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Notice in the previous example that the simplified form of is , which is the product of an integer and a square root. We always write the integer in front of the square root.
- Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
- Use the product rule to rewrite the radical as the product of two radicals.
- Simplify the square root of the perfect square.
EXAMPLE 2
Simplify: .
| Rewrite the radicand as a product using the largest perfect square factor. | |
| Rewrite the radical as the product of two radicals. | |
| Simplify. |
TRY IT 2.1
Simplify: .
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TRY IT 2.2
Simplify: .
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We could use the simplified form to estimate . We know 5 is between 2 and 3, and is . So is between 20 and 30.
The next example is much like the previous examples, but with variables.
EXAMPLE 3
Simplify: where .
| Rewrite the radicand as a product using the largest perfect square factor. | |
| Rewrite the radical as the product of two radicals. | |
| Simplify. |
TRY IT 3.1
Simplify: where .
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TRY IT 3.2
Simplify: where .
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We follow the same procedure when there is a coefficient in the radical, too.
EXAMPLE 4
Si