Square Roots Practice Worksheet - Derry Area School District

[Pages:10]9.2 Simplifying Radical Expressions

9.2 OBJECTIVES 1. Simplify expressions involving numeric radicals 2. Simplify expressions involving algebraic radicals

In Section 9.1, we introduced the radical notation. For most applications, we will want to make sure that all radical expressions are in simplest form. To accomplish this, the following three conditions must be satisfied.

Rules and Properties: Square Root Expressions in Simplest Form

An expression involving square roots is in simplest form if 1. There are no perfect-square factors in a radical. 2. No fraction appears inside a radical. 3. No radical appears in the denominator.

For instance, considering condition 1, 117 is in simplest form because 17 has no perfect-square factors whereas 112 is not in simplest form because it does contain a perfect-square factor. 112 14 3

A perfect square

To simplify radical expressions, we'll need to develop two important properties. First, look at the following expressions:

14 9 136 6 14 19 2 3 6 Because this tells us that 14 9 14 19, the following general rule for radicals is suggested.

Rules and Properties: Property 1 of Radicals For any positive real numbers a and b, 1ab 1a 1b In words, the square root of a product is the product of the square roots.

707

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CHAPTER 9 EXPONENTS AND RADICALS

Let's see how this property is applied in simplifying expressions when radicals are involved.

Example 1 Simplifying Radical Expressions

NOTE Perfect-square factors are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on.

Simplify each expression. (a) 112 14 3

NOTE Apply Property 1.

NOTE Notice that we have removed the perfect-square factor from inside the radical, so the expression is in simplest form. NOTE It would not have helped to write 145 115 3 because neither factor is a perfect square. NOTE We look for the largest perfect-square factor, here 36.

NOTE Then apply Property 1.

A perfect square

14 13 213 (b) 145 19 5

A perfect square

19 15 315 (c) 172 136 2

A perfect square

136 12 6 12 (d) 5118 519 2

A perfect square

5 19 12 5 3 12 1512

CAUTION

Be Careful! Even though

1a b 1a 1b 1a b is not the same as

1a 1b

Let a 4 and b 9, and substitute.

1a b 14 9 113 1a 1b 14 19 2 3 5

Because 113 5, we see that the expressions 1a b and 1a 1b are not in general the same.

CHECK YOURSELF 1 Simplify.

(a) 120

(b) 175

(c) 198

(d) 148

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SIMPLIFYING RADICAL EXPRESSIONS SECTION 9.2

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The process is the same if variables are involved in a radical expression. In our remaining work with radicals, we will assume that all variables represent positive real numbers.

Example 2 Simplifying Radical Expressions Simplify each of the following radicals. (a) 2x3 2x2 x

NOTE By our first rule for radicals.

NOTE 2x2 x (as long as x is positive).

A perfect square

2x2 1x x1x

(b) 24b3 24 b2 b

Perfect squares

24b2 1b 2b1b

NOTE Notice that we want the perfect-square factor to have the largest possible even exponent, here 4. Keep in mind that

a2 a2 a4

(c) 218a5 29 a4 2a

Perfect squares

29a4 12a 3a2 12a

CHECK YOURSELF 2 Simplify.

(a) 29x3

(b) 227m3

(c) 250b5

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To develop a second property for radicals, look at the following expressions:

16 A4

14

2

116 14

4 2

2

16 Because A 4

116 14 , a second general rule for radicals is suggested.

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CHAPTER 9 EXPONENTS AND RADICALS

Rules and Properties: Property 2 of Radicals

For any positive real numbers a and b,

a Ab

1a 1b

In words, the square root of a quotient is the quotient of the square roots.

This property is used in a fashion similar to Property 1 in simplifying radical expressions. Remember that our second condition for a radical expression to be in simplest form states that no fraction should appear inside a radical. Example 3 illustrates how expressions that violate that condition are simplified.

NOTE Apply Property 2 to write the numerator and denominator as separate radicals.

NOTE Apply Property 2.

NOTE Apply Property 2. NOTE Factor 8x2 as 4x2 2.

NOTE Apply Property 1 in the numerator.

Example 3 Simplifying Radical Expressions

Write each expression in simplest form.

(a)

9 A4

19 14

3 2

Remove any

perfect squares from the radical.

(b)

2 A 25

12 125

12 5

(c)

8x2 B9

28x2 19

24x2 2 3

24x2 12 3

2x12 3

CHECK YOURSELF 3 Simplify.

25 (a) A 16

7 (b) A 9

12x2 (c) B 49

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SIMPLIFYING RADICAL EXPRESSIONS SECTION 9.2

711

In our previous examples, the denominator of the fraction appearing in the radical was a perfect square, and we were able to write each expression in simplest radical form by removing that perfect square from the denominator.

If the denominator of the fraction in the radical is not a perfect square, we can still apply Property 2 of radicals. As we will see in Example 4, the third condition for a radical to be in simplest form is then violated, and a new technique is necessary.

Example 4 Simplifying Radical Expressions Write each expression in simplest form.

NOTE We begin by applying Property 2.

(a)

1 A3

11 13

1 13

1 Do you see that 13 is still not in simplest form because of the radical in the denominator? To solve this problem, we multiply the numerator and denominator by 13. Note that the denominator will become

13 13 19 3

We then have

NOTE We can do this because we are multiplying the fraction

13 by 13 or 1, which does not change its value.

NOTE 12 15 12 5 110 15 15 5

1 13

1 13 13 13

13 3

13 The expression is now in simplest form because all three of our conditions are satisfied.

3

(b)

2 A5

12 15

12 15

15 15

110 5

and the expression is in simplest form because again our three conditions are satisfied.

NOTE We multiply numerator and denominator by 17 to "clear" the denominator of the radical. This is also known as "rationalizing" the denominator.

(c)

3x A7

13x 17

13x 17 17 17

121x 7

The expression is in simplest form.

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CHAPTER 9 EXPONENTS AND RADICALS

CHECK YOURSELF 4 Simplify.

1 (a) A 2

2 (b) A 3

2y (c) A 5

Both of the properties of radicals given in this section are true for cube roots, fourth roots, and so on. Here we have limited ourselves to simplifying expressions involving square roots.

CHECK YOURSELF ANSWERS 1. (a) 215; (b) 513; (c) 712; (d) 413

2. (a) 3x1x; (b) 3m13m;

(c) 5b2 12b

5 17 2x13

3. (a) ; (b) ; (c)

43

7

12 16 110y

4. (a) ; (b) ; (c)

2

3

5

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9.2 Exercises

Use Property 1 to simplify each of the following radical expressions. Assume that all variables represent positive real numbers.

1. 118

2. 150

3. 128

4. 1108

5. 145

6. 180

7. 148

8. 1125

9. 1200

10. 196

11. 1147

12. 1300

13. 3 112

14. 5 124

15. 25x2

16. 27a2

17. 23y4

18. 210x6

19. 22r3

20. 25a5

21. 227b2

22. 298m4

23. 224x4

24. 272x3

Name Section

Date

ANSWERS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14 15. 16. 17. 18. 19. 20. 21. 22. 23. 24

713

ANSWERS 25. 26. 27. 28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

714

25. 254a5

26. 2200y6

27. 2x3y2

28. 2a2b5

Use Property 2 to simplify each of the following radical expressions.

4 29. A 25

64 30. A 9

9 31. A 16

49 32. A 25

3 33. A 4

5 34. A 9

5 35. A 36

10 36. A 49

Use the properties for radicals to simplify each of the following expressions. Assume that all variables represent positive real numbers.

8a2 37.

B 25

12y2 38.

B 49

1 39. A 5

1 40. A 7

3 41. A 2

5 42. A 3

3a 43. A 5

2x2 45. B 3

8s3 47.

B7

2x 44. A 7

5m2 46. B 2

12x3 48.

B5

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