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Simplify: 75 x 5 3 x .

5 x 2

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Simplify: 72 z 12 2 z 10 .

6 z

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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.

( a b ) m = a m b m , b 0

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.

Quotient property of square roots

If a , b are non-negative real numbers and b 0 , then

a b = a b

Simplify: 21 64 .

Solution

21 64 We cannot simplify the fraction inside the radical. Rewrite using the quotient property. 21 64 Simplify the square root of 64. The numerator cannot be simplified. 21 8

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How to use the quotient property to simplify a square root

Simplify: 27 m 3 196 .

Solution

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.
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Simplify: 24 p 3 49 .

2 p 6 p 7

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Simplify: 48 x 5 100 .

2 x 2 3 x 5

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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.

Simplify: 45 x 5 y 4 .

Solution

45 x 5 y 4 We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property. 45 x 5 y 4 Simplify the radicals in the numerator and the denominator. 9 x 4 · 5 x y 2 Simplify. 3 x 2 5 x y 2

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Simplify: 80 m 3 n 6 .

4 m 5 m n 3

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Simplify: 54 u 7 v 8 .

3 u 3 6 u v 4

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

Simplify: 81 d 9 25 d 4 .

Solution

81 d 9 25 d 4 Simplify the fraction in the radicand. 81 d 5 25 Rewrite using the Quotient Property. 81 d 5 25 Simplify the radicals in the numerator and the denominator. 81 d 4 · d 5 Simplify. 9 d 2 d 5

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Simplify: 64 x 7 9 x 3 .

8 x 2 3

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Simplify: 16 a 9 100 a 5 .

2 a 2 5

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Simplify: 18 p 5 q 7 32 p q 2 .

Solution

18 p 5 q 7 32 p q 2 Simplify the fraction in the radicand, if possible. 9 p 4 q 5 16 Rewrite using the Quotient Property. 9 p 4 q 5 16 Simplify the radicals in the numerator and the denominator. 9 p 4 q 4 · q 4 Simplify. 3 p 2 q 2 q 4

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Simplify: 50 x 5 y 3 72 x 4 y .

5 y x 6

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Simplify: 48 m 7 n 2 125 m 5 n 9 .

4 m 3 5 n 3 5 n

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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
    a b = 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 b 0 , then
    a b = a b



  • 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.

Practice makes perfect

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

147 m 7 n 11

7 m 3 n 5 3 m n

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75 r 13 s 9

5 r 6 s 4 3 r s 70)

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300 p 9 q 11

10 p 4 q 5 3 p q

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242 m 13 n 21

11 m 6 n 10 2 m n

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Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

75 r 9 s 8

5 r 4 3 r s 4

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32 x 5 y 3 18 x 3 y

4 x y 3

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27 p 2 q 108 p 5 q 3

1 2 p q p

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Everyday math

  1. Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify 288 to determine the length and the width of his garden. Round to the nearest tenth of a foot.
  2. Suppose Elliott decides to reduce the size of his square garden so that he can create a 5-foot-wide walking path on the north and east sides of the garden. Simplify 288 5 to determine the length and width of the new garden. Round to the nearest tenth of a foot.

17.0 feet 15.0 feet

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  1. Melissa accidentally drops a pair of sunglasses from the top of a roller coaster, 64 feet above the ground. Simplify 64 16 to determine the number of seconds it takes for the sunglasses to reach the ground.
  2. Suppose the sunglasses in the previous example were dropped from a height of 144 feet. Simplify 144 16 to determine the number of seconds it takes for the sunglasses to reach the ground.
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Writing exercises

Explain why x 4 = x 2 . Then explain why x 16 = x 8 .

Answers will vary.

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Explain why 7 + 9 is not equal to 7 + 9 .

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Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and three rows. The columns are labeled, “I can…,” “confidently,” “with some help,” and “no—I don’t get it!” The rows under “I can…” Read, “use the Product Property to simplify square roots.,” and “use the Quotient Property to simplify square roots.” The other rows unders the other columns are blank.

After reviewing this checklist, what will you do to become confident for all objectives?

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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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