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9.2: Simplify Square Roots

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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
BE PREPARED

Before you get started take this readiness quiz.

  1. Simplify: 80176.
    If you missed this problem, review [link].
  2. Simplify: n9n3.
    If you missed this problem, review [link].
  3. Simplify: q4q12.
    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.

Definition: 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 (ab)m=ambm. The corresponding property of square roots says that ab=a·b.

Definition: PRODUCT PROPERTY OF SQUARE ROOTS

If a, b are non-negative real numbers, then ab=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 Example.

How To Use the Product Property to Simplify a Square Root

Example 9.2.1

Simplify: 50.

Answer

This figure has three columns and three rows. The first row says, “Step 1. Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.” It then says, “25 is the largest perfect square factor of 50. 50 equals 25 times 2. Always write the perfect square factor first.” Then it shows the square root of 50 and the square root of 25 times 2.The second row says, “Step 2. Use the product rule to rewrite the radical as the product of two radicals.” The second column is empty, but the third column shows the square root of 25 times the square root of 2.The third row says, “Step 3. Simplify the square root of the perfect square.” The second column is empty, but the third column shows 5 times the square root of 2.

Example 9.2.2

Simplify: 48.

Answer

43

Example 9.2.3

Simplify: 45.

Answer

35

Notice in the previous example that the simplified form of 50 is 52, which is the product of an integer and a square root. We always write the integer in front of the square root.

Definition: 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.
Example 9.2.4

Simplify: 500.

Answer

500Rewrite the radicand as a product using the largest perfect square factor100·5Rewrite the radical as the product of two radicals100·5Simplify105

Example 9.2.5

Simplify: 288.

Answer

122

Example 9.2.6

Simplify:432.

Answer

123

We could use the simplified form 105 to estimate 500. We know 5 is between 2 and 3, and 500 is 105. So 500 is between 20 and 30.

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

Example 9.2.7

Simplify: x3.

Answer

x3Rewrite the radicand as a product using the largest perfect square factorx2·xRewrite the radical as the product of two radicalsx2·xSimplifyxx

Example 9.2.8

Simplify:b5.

Answer

b2b

Example 9.2.9

Simplify: p9.

Answer

p4p

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

Example 9.2.10

Simplify: 25y5.

Answer

25y5Rewrite the radicand as a product using the largest perfect square factor.25y4·yRewrite the radical as the product of two radicals.25y4·ySimplify.5y2y

Example 9.2.11

Simplify: 16x7.

Answer

4x3x

Example 9.2.12

Simplify: 49v9.

Answer

7v4v

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

Example 9.2.13

Simplify: 72n7.

Answer

72n7Rewrite the radicand as a product using the largest perfect square factor.36n6·2nRewrite the radical as the product of two radicals.36n6·2nSimplify.6n32n

Example 9.2.14

Simplify: 32y5.

Answer

4y22y

Example 9.2.15

Simplify: 75a9.

Answer

5a43a

Example 9.2.16

Simplify: 63u3v5.

Answer

63u3v5Rewrite the radicand as a product using the largest perfect square factor.9u2v4·7uvRewrite the radical as the product of two radicals.9u2v4·7uvSimplify.3uv27uv

Example 9.2.17

Simplify: 98a7b5.

Answer

7a3b22ab

Example 9.2.18

Simplify: 180m9n11.

Answer

6m4n55mn

We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify 25+144 we must simplify each square root separately first, then add to get the sum of 17.

The expression 17+7 cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.

In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.

Example 9.2.19

Simplify: 3+32.

Answer

3+32Rewrite the radicand as a product using the largest perfect square factor.3+16·2Rewrite the radical as the product of two radicals.3+16·2Simplify.3+42

The terms are not like and so we cannot add them. Trying to add an integer and a radical is like trying to add an integer and a variable—they are not like terms!

Example 9.2.20

Simplify: 5+75.

Answer

5+53

Example 9.2.21

Simplify: 2+98.

Answer

2+72

The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

Example 9.2.22

Simplify: 4482.

Answer

4482Rewrite the radicand as a product using thelargest perfect square factor.416·32Rewrite the radical as the product of two radicals.416·32Simplify.4432Factor the common factor from thenumerator.4(13)2Remove the common factor, 2, from thenumerator and denominator.2(13)

Example 9.2.23

Simplify: 10755.

Answer

23

Example 9.2.24

Simplify: 6453.

Answer

25

Use the Quotient Property to Simplify Square Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.

Example 9.2.25

Simplify: 964.

Answer

964Since(38)238

Example 9.2.26

Simplify: 2516.

Answer

54

Example 9.2.27

Simplify: 4981.

Answer

79

If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!

Example 9.2.28

Simplify: 4580.

Answer

4580Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator.5·95·16Simplify the fraction by removing common factors.916Simplify.(34)2=91634

Example 9.2.29

Simplify: 7548.

Answer

54

Example 9.2.30

Simplify: 98162.

Answer

79

In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, aman=amn, a0.

Example 9.2.31

Simplify: m6m4.

Answer

m6m4Simplify the fraction inside the radical firstm2Divide the like bases by subtracting the exponents.Simplify.m

Example 9.2.32

Simplify: a8a6.

Answer

a

Example 9.2.33

Simplify: x14x10.

Answer

x2

Example 9.2.34

Simplify: 48p73p3.

Answer

48p73p3Simplify the fraction inside the radical first.16p4Simplify.4p2

Example 9.2.35

Simplify: 75x53x.

Answer

5x2

Example 9.2.36

Simplify: 72z122z10.

Answer

6z

Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.

(ab)m=ambm, b0

We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.

Definition: QUOTIENT PROPERTY OF SQUARE ROOTS

If a, b are non-negative real numbers and b0, then

ab=ab

Example 9.2.37

Simplify: 2164.

Answer

2164We cannot simplify the fraction inside the radical. Rewrite using the quotient property.2164Simplify the square root of 64. The numerator cannot be simplified.218

Example 9.2.38

Simplify: 1949.

Answer

197

Example 9.2.39

Simplify:2881

Answer

279

How to Use the Quotient Property to Simplify a Square Root

Example 9.2.40

Simplify: 27m3196.

Answer

This table has three columns and three rows. The first row reads, “Step 1. Simplify the fraction in the radicand, if possible.” Then it shows that 27 m cubed over 196 cannot be simplified. Then it shows the square root of 27 m cubed over 196.The second row says, “Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.” Then it says, “We rewrite the square root of 27 m cubed over 196 as the quotient of the square root of 27 m cubed and the square root of 196.” Then it shows the square root of 27 m cubed over the square root of 196.The third row says, “Step 3. Simplify the radicals in the numerator and the denominator.” Then it says, “9 m squared and 196 are perfect squares.” It then shows the square root of 9 m squared time the square root of 3 m over the square root of 196. It then shows 3 m times the square root of 3 m over 14.

Example 9.2.41

Simplify: 24p349

Answer

2p6p7

Example 9.2.42

Simplify: 48x5100

Answer

2x23x5

Definition: SIMPLIFY A SQUARE ROOT USING THE QUOTIENT PROPERTY.
  1. Simplify the fraction in the radicand, if possible.
  2. Use the Quotient Property to rewrite the radical as the quotient of two radicals.
  3. Simplify the radicals in the numerator and the denominator.
Example 9.2.43

Simplify: 45x5y4.

Answer

45x5y4We cannot simplify the fraction inside the radical. Rewrite using the quotient property.45x5y4Simplify the radicals in the numerator and the denominator.9x45xy2Simplify.3x25xy2

Example 9.2.44

Simplify: 80m3n6

Answer

4m5mn3

Example 9.2.45

Simplify: 54u7v8.

Answer

3u36uv4

Be sure to simplify the fraction in the radicand first, if possible.

Example 9.2.46

Simplify: 81d925d4.

Answer

81d925d4Simplify the fraction in the radicand.81d525Rewrite using the quotient property.81d525Simplify the radicals in the numerator and the denominator.81d4d5Simplify.9d2d5

Example 9.2.47

Simplify: 64x79x3.

Answer

8x23

Example 9.2.48

Simplify: 16a9100a5.

Answer

2a25

Example 9.2.49

Simplify: 18p5q732pq2.

Answer

18p5q732pq2Simplify the fraction in the radicand.9p4q516Rewrite using the quotient property.9p4q516Simplify the radicals in the numerator and the denominator.9p4q4q4Simplify.3p2q2q4

ExAMPLe 9.2.50

Simplify: 50x5y372x4y.

Answer

5yx6

Example 9.2.51

Simplify: 48m7n2125m5n9.

Answer

4m35n35n

Key Concepts

  • Simplified Square Root a is considered simplified if a has no perfect-square factors.
  • Product Property of Square Roots If a, b are non-negative real numbers, then

    ab=a·b

  • Simplify a Square Root Using the Product Property To 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.
  • Quotient Property of Square Roots If a, b are non-negative real numbers and b0, then

    ab=ab

  • Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
    1. Simplify the fraction in the radicand, if possible.
    2. Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
    3. Simplify the radicals in the numerator and the denominator.

This page titled 9.2: Simplify Square Roots is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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