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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

Before you get started take this readiness quiz.

  1. Simplify: 80 176 .
    If you missed this problem, review [link] .
  2. Simplify: n 9 n 3 .
    If you missed this problem, review [link] .
  3. Simplify: q 4 q 12 .
    If you missed this problem, review [link] .

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that 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 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.

Simplified square root

a is considered simplified if a has no perfect square factors.

So 31 is simplified. But 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 ( a b ) m = a m b m . The corresponding property of square roots says that a b = a · b .

Product property of square roots

If a , b are non-negative real numbers, then a b = a · 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

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

Simplify a square root using the product property.

  1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect-square factor.
  2. Use the product rule to rewrite the radical as the product of two radicals.
  3. Simplify the square root of the perfect square.

Simplify: 500 .

Solution

500 Rewrite the radicand as a product using the largest perfect square factor. 100 · 5 Rewrite the radical as the product of two radicals. 100 · 5 Simplify. 10 5

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

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

Simplify: x 3 .

Solution

x 3 Rewrite the radicand as a product using the largest perfect square factor. x 2 · x Rewrite the radical as the product of two radicals. x 2 · x Simplify. x x

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

Simplify: 25 y 5 .

Solution

25 y 5 Rewrite the radicand as a product using the largest perfect square factor. 25 y 4 · y Rewrite the radical as the product of two radicals. 25 y 4 · y Simplify. 5 y 2 y

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Simplify: 16 x 7 .

4 x 3 x

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Simplify: 49 v 9 .

7 v 4 v

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

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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