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
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 4.1
Simplify: where .
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TRY IT 4.2
Simplify: where .
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In the next example both the constant and the variable have perfect square factors.
EXAMPLE 5
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 5.1
Simplify: where .
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TRY IT 5.2
Simplify: .
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EXAMPLE 6
Simplify: where and .
| Rewrite the radicand as a product using the largest perfect square factor. | |
| Rewrite the radical as the product of two radicals. | |
| Simplify. |
TRY IT 6.1
Simplify: where and .
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TRY IT 6.2
Simplify: where .
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We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify we must simplify each square root separately first, then add to get the sum of 17
The expression 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 7
Simplify: .
| Rewrite the radicand as a product using the largest perfect square factor. | |
| Rewrite the radical as the product of two radicals. | |
| Simplify. |
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!
TRY IT 7.1
Simplify: .
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TRY IT 7.2
Simplify: .
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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 8
Simplify: .
| Rewrite the radicand as a product using the largest perfect square factor. | |
| Rewrite the radical as the product of two radicals. | |
| Simplify. | |
| Factor the common factor from the numerator. | |
| Remove the common factor, 2, from the numerator and denominator. | because |
| Simplify. |
TRY IT 8.1
Simplify: .
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TRY IT 8.2
Simplify: .
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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
Simplify: .
TRY IT 9.1
Simplify: .
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TRY IT 9.2
Simplify: .
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If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!
EXAMPLE 10
Simplify: .
| Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator. | |
| Simplify the fraction by removing common factors. | |
TRY IT 10.1
Simplify: .
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TRY IT 10.2
Simplify: .
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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, .
EXAMPLE 11
Simplify: where .
| Simplify the fraction inside the radical first. Divide the like bases by subtracting the exponents. | |
| Simplify. |
TRY IT 11.1
Simplify: where .
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TRY IT 11.2
Simplify: where .
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EXAMPLE 12
Simplify: where .
| Simplify the fraction inside the radical first. | |
| Simplify. |
TRY IT 12.1
Simplify: where .
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TRY IT 12.2
Simplify: where .
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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.
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 , then
EXAMPLE 13
Simplify: .
| We cannot simplify the fraction inside the radical. Rewrite using the quotient property. | |
| Simplify the square root of 64. The numerator cannot be simplified. |
TRY IT 13.1
Simplify: .
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TRY IT 13.2
Simplify: .
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EXAMPLE 14
Simplify: where .
| Step 1. Simplify the fraction in the radicand, if possible. | cannot be simplified. | |
| Step 2. Use the Quotient Property to rewrite the radical as the quotient of two radicals. |
We rewrite as the quotient of and . |
|
| Step 3. Simplify the radicals in the numerator and the denominator. | We know that and and are perfect squares. |
TRY IT 14.1
Simplify: where .
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TRY IT 14.2
Simplify: where .
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- Simplify the fraction in the radicand, if possible.
- Use the Quotient Property to rewrite the radical as the quotient of two radicals.
- Simplify the radicals in the numerator and the denominator.
EXAMPLE 15
Simplify: where and .
| We cannot simplify the fraction in the radicand. Rewrite using the Quotient Property. | |
| Simplify the radicals in the numerator and the denominator. | |
| Simplify. |
TRY IT 15.1
Simplify: where and .
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TRY IT 15.2
Simplify: where and .
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Be sure to simplify the fraction in the radicand first, if possible.
EXAMPLE 16
Simplify: where .
| Simplify the fraction in the radicand. | |
| Rewrite using the Quotient Property. | |
| Simplify the radicals in the numerator and the denominator. | |
| Simplify. |
TRY IT 16.1
Simplify: where .
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TRY IT 16.2
Simplify: where .
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EXAMPLE 17
Simplify: where and .
| Simplify the fraction in the radicand, if possible. | |
| Rewrite using the Quotient Property. | |
| Simplify the radicals in the numerator and the denominator. | |
| Simplify. |
TRY IT 17.1
Simplify: where and .
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TRY IT 17.2
Simplify: where and .
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Key Concepts
- Simplified Square Root is considered simplified if has no perfect-square factors.
- Product Property of Square Roots If a, b are non-negative real numbers, then
- Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:
- 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.
- Quotient Property of Square Roots If a, b are non-negative real numbers and , then
- Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
- Simplify the fraction in the radicand, if possible.
- Use the Quotient Rule to rewrite the radical as the quotient of two radicals.
- 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.
| 1. | 2. |
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| 41. | 42. |
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| 45. | 46. |
| 47. | 48. |
Use the Quotient Property to Simplify Square Roots
In the following exercises, simplify. Note that all the variables are non-negative.
| 49. | 50. |
| 51. | 52. |
| 53. | 54. |
| 55. | 56. |
| 57. | 58. |
| 59. | 60. |
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| 85. | 86. |
| 87. | 88. |
Everyday Math
| 89.
a) Elliott decides to construct a square garden that will take up 288 square feet of his yard. Simplify to determine the length and the width of his garden. Round to the nearest tenth of a foot. b) 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 to determine the length and width of the new garden. Round to the nearest tenth of a foot. |
90.
a) Melissa accidentally drops a pair of sunglasses from the top of a roller coaster, 64 feet above the ground. Simplify to determine the number of seconds it takes for the sunglasses to reach the ground. b) Suppose the sunglasses in the previous example were dropped from a height of 144 feet. Simplify to determine the number of seconds it takes for the sunglasses to reach the ground. |
Writing Exercises
| 91. Explain why . Then explain why . | 92. Explain why is not equal to . |
Answers
| 1. | 3. | 5. |
| 7. | 9. | 11. |
| 13. | 15. | 17. |
| 19. | 21. | 23. |
| 25. | 27. | 29. |
| 31. | 33. | 35. |
| 37. | 39. | 41. |
| 43. | 45. | 47. |
| 49. | 51. | 53. |
| 55. | 57. | 59. |
| 61. | 63. | 65. |
| 67. | 69. | 71. |
| 73. | 75. | 77. |
| 79. | 81. | 83. |
| 85. | 87. | 89. a) b) |
| 91. Answers will vary. |
Attributions
This chapter has been adapted from “Simplify Square Roots” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.