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Name:__________________________________________ ALSO PRINT YOUR NAME AT TOP OF COVER PAGE OF UNIT. Videos produced on YouTube by YourMathGal Julie Harland that accompany sections in this Unit are at https://sites.google.com/site/packet2int/home/p2videos To navigate to that site, go to YourMathGal.com. Click Home. Click My Books on side bar. Click Math 64. Click Unit 1. Click Videos for Unit 2. Before doing the problems in each section, watch and transcribe the videos accompanying that section for your notes. *An asterisk next to a problem indicates its solution is worked out on a video. Intermediate Algebra Unit 2 Radicals, Rational Exponents, Complex Numbers 2.1 Square Roots .................................................................................................... 2 2.2 Higher Order Roots .................................................................................... 12 2.3 Add and Multiply Radicals ....................................................................... 16 2.4 Divide Radicals ............................................................................................. 27 2.5 Rational Exponents .................................................................................... 41 2.6 Radical Equations ........................................................................................ 54 2.7 Pythagorean Theorem & Distance Formula .................................... 64 2.8 Complex Numbers ...................................................................................... 72 2.9 Dividing Complex Numbers .................................................................... 85 2.10 Review Exercises ......................................................................................... 93 ISBN: 978-‐0-‐8401-‐3255-‐0 Printing Date: February 2016 This Unit is (c) 2015 Julie Harland and is licensed under a Creative Commons Attribution-‐NonCommercial-‐ShareAlike 4.0 International License

of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra Intermediate Algebra

Print First and Last Name:_____________________________________________________________ 2.1 Square Roots 1. Evaluate the following:

*a. 02 = *b. 32 = *c. −2( )2 =

*d. 56

⎛⎝⎜

⎞⎠⎟2

= *e. 30( )2 = f. −40( )2 =

*g. .4( )2 = h. .07( )2 = *i. −3( )2 =

j. −5( )2 = k. 52 = l. 202 = 2. State the two square roots of each number. *a. 16: _____________ b. 100: _____________ 3. What symbol is used to denote the “principal” square root of a number and what

does “principal square root of a number” mean? 4. Simplify. If it’s not a real number, write NOT REAL. Each problem has one answer.

*a. 16 = *b. 81 = *c. 36 =

d. 0 = e. 1 = *f. −9 =

*g. − 49 = h.

100121

= *i.

916

=

j. 2500 = k. − 64 = l. −49 =


Square Roots with variables Simplify . Assume all variables are positive.

*5. x2( )2 = *6.

x3( )2 =

*7. x8( )2 = 8. x5 ⋅ x5 =

*9. x4 = *10. n16 =

*11. x6 = *12. x16 =

13. m8 = 14. m12 =

*15. b30 = 16. y22 =

*17. 16x16 = *18. 36m

10 =

*19. 9m8n4y2 =

21. 25m24n6 =

23. 81a100b60c40 =

24. Give 3 examples of irrational numbers involving square roots.


Approximating Square Roots 25. List the perfect squares from 22 up to 122 . Start with 4, 9, etc. ________________________________________________________________________________________________ Use a calculator to approximate each square root to the 3 places after the decimal point. Then square the approximation and round to 5 places after the decimal point.

*26. 15 ≈ ________________ Square that approximation:_____________________________

*27. 30 ≈ ________________ Square that approximation: _____________________________

28. 41 ≈ ________________ Square that approximation: _____________________________

*29. Enter 3 in a calculator, round to 3 places after the decimal point and write on the first blank below. Then multiply by 2. Then enter 2 3 in a calculator and round.

a. 2 3( ) ≈ 2 · ________________ = __________________ b. 2 3 ≈ ____________________

30. Enter 5 in a calculator, round to 3 places after the decimal point and write on the first blank below. Then multiply by 3. Then enter 3 5 in a calculator and round.

a. 3 5( ) ≈ 3 · ________________ = __________________ b. 3 5 ≈ ____________________


Multiplication Property of Square Roots *31. Complete the following statements.

a. If x ≥ 0, then x2 = _______ b. x x = ___ c. x( )2 = ___ 32. What is the main difference between part a above compared to part b and c? 33. Evaluate the following. All of these are in the form shown in b or c above.

*a. 7 7 = *b. 103 103 =

c. 65 65 = *d. junk junk =

*e. 5x3y 5x3y = f. 2x − 5 2x − 5 =

*g. stuff( )2 =

*h. 19( )2 =

i. m3 + 5n( )2 = j. 101( )2 =

k. name( )2 =

l. math math = Even though the square root of a negative number is not a real number, you can still compute the following using part b and c from question 1 to simplify these. Your answer should be a negative number. We’ll cover square roots of negative numbers later.

m. −7( )2 =

n. −14 −14 =


MULTIPLICATION PROPERTY FOR SQUARE ROOTS

If a ≥ 0 and b ≥ 0 , then a ⋅ b = ab To simplify things and make directions in this unit less cumbersome, we will make the assumption that all variables in this unit are positive unless otherwise stated, and write the

multiplication property as a ⋅ b = ab Multiply. Then simplify if possible. Assume variables are positive.

*34. 3 ⋅ 7 = *35. 2 ⋅ 8 =

*36. 5 ⋅ 5 = *37. 47 ⋅ 47 =

*38. 3 ⋅ 5 = 39. 11 ⋅ 2 =

*40. a ⋅ a5 = 41. x3 ⋅ x5 =

*42. 2 ⋅ 3 ⋅ 7 = 43. 5 ⋅ 3 ⋅ 2 =

*44. 2x3 ⋅ 18x5 = 45. 2m

7 ⋅ 8m9 =

*46. 23x5 ⋅ 23x7 =

47. 17ab3 ⋅ 17a7b =

48. 27x5y15 ⋅ 3x9y =

49. x3 2x5 ⋅3 8x9 =


Simplifying Square Roots A square root of a counting number is considered simplified if no factors greater than 1 are perfect squares. The number under the square root symbol is called the radicand. If the radicand has a perfect square factor, the square root of that number can be simplified. 50. List all factors for each radicand, and circle any perfect squares greater than 1. Then state if the given square root is simplifed

*a. 15 : State the factors of 15: ______________________________________________________

Is 15 simplified? ___________

*b. 12 : State the factors of 12: ______________________________________________________


c. 30 : State the factors of 30: ______________________________________________________


d. 45 : State the factors of 45: ______________________________________________________


*51. Show all steps to simplify 12


Show steps to simplify. Use a calculator to verify the approximation of the original square root and the simplified answer are the same. Put a BOX around each answer.

*52. 18

53. 75

*54. 50

55. 40

*56. 54

*57. 72

58. 28

59. 99

*60. 3 75

61. 5 24

62. 2 27


Show steps to simplify. . Assume variables are positive Put a BOX around each answer.

*63. x15

*64. m11

65. n33

66. a2m7

67. x13y6

68. 44m11

*69. 48x8n5

70. 98m10n3

71. 22a9n11

of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra

*72. −6 45x3

*73. 4x 28x2

74. 3m 32m5

*75. 4x2m 63x2m7

*76. −5x4 8x2y9

77. x3n 18x8n15

78. −5x3n 20x5n7

of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra

Multiply and simplify. Show Steps. Assume variables are positive. Put BOX around answer.

*79. 21 ⋅ 14

*80. 35 ⋅ 55

81. 30 ⋅ 42

82. 30 ⋅ 33

83. 22x3 33x

*84. 6x3 ⋅ 2x5 ⋅ 5x7

*85. 12x3 ⋅ 3x

*86. −3x2 10 ⋅5 30

87. −5x2 6x4 ⋅2x3 10x5

of 98 Unit 2 Harland 2.2 Higher Order Roots Intermediate Algebra

2.2 Higher Order Roots

If a ≥ 0 , then ann = a Simplify. If it’s not a real number, write “not a real number”. Assume variables are positive. Show all steps starting with #16. Pay close attention to the index, which is the little number you see on the left (except on the square root, where it is optional to write the index of 2). So for #1, the index is 3, which means you are taking the cube root of 64.

*1. 643 2. 814

*3. 325

4. −273 *5. 100004

9. −16

*10. m8( )3 *11. m243 12. x183

13. n505 14. n147

15. n124

*16. 403

*17. 543

18. 804

19. 2503

*20. m143

21. x135


22. n314

*23. 563

*24. x103

*25. 24x103

*26. x263

27. x10m125

28. −3x 4x143

*29. 80x12y154

30. 5xy −40x20y193


We have been using this definition: If a ≥ 0 , then ann = a But what if we are not sure whether a is positive or negative? It depends on the index, n. Below is the definition which is true regardless of the sign of a.

If n is an even integer, then ann = a If n is an odd integer, then ann = a Simplify. Use absolute value signs as needed. Do not assume variables are positive. Do not leave negative numbers under the radical sign. Write “NOT REAL if it is not a real number. BOX answer.

*31. b33 *32. m44

33. n99 34. m66

*35. 164 *36. −83

37. −3( )44 38. −19

*39. −16 40*. −814

41. −1 42. 766

*43. −1253 *44. −23

45. m20n84 46. m20n305


Simplify. Use absolute value signs as needed. Do not assume variables are positive. Do not leave negative numbers under the radical sign. BOX answer.

*47. −250x6y103

48. −32x11y155

*49. 45x3y9

50. −63x26y93

51. 48x10y154

52. 64x12y155

of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra

2.3 Add and Multiply Radicals Simplify if possible. If already simplified, write “SIMPLIFIED”. Assume variables are positive. Show all steps starting with #13. BOX answer.

*1. 3c + 5c *2. 3 2 + 5 2

*3. 3 7 + 5 7 4. 6x + x

5. 6 5 + 5 *6. 2 3 + 5 2

7. 6 5m − 3 5m 8. b −10b

9. 2 −10 2 10. 2 2x − 3 2x

11. 7 3y − 2 3y + 3y

12. 7 3n − 2 n + 3n

*13. 8 + 18

14. 12 + 27

15. 2 27 + 3 50

16. 4 75 − 50


*17. 5x 45x3 − 3 20x5

18. 5 8x + 2 18x − 50x

*19. 3m 18x2 − x 27m2

20. 3 12x2 − 2x 27

*21. 4x 32 − 18x2 + 2x 128


22. 8x 75m + 3 50m

23. 7 20x5 − x 24x3

24. 3m 45x8 − x3 54x2m2

25. 3m 28 − 20 + 44m2


Adding Higher Order Roots Simplify if possible. Assume variables are positive. Show all steps. BOX answer.

26. 8 73 + 4 73

27. 135 − 4 135

28. 24 + 5 23 − 3 2

29. 9 163 + 3 23

*30. 5 543 + 3 2503

31. 3 323 + 5 1083

*32. 18 + 503

33. 98 + 163


*34. 10 403 − 5 6253

35. 5 563 − 70003

*36. 8 x94 − x2 81x4

37. 8 x135 − x 16x74

*38.

995x

− 44x2

39.

455x2

− 80x4


Multiplying Binomials with Radicals Use the distributive Property to Multiply and Simplify. Show all steps. BOX answer.

40. 2 2 + 3( )

41. 5 4 + 5( )

42. 7 2 7 + 3( )

43. 5 8 2 − 3 5( )

44. 2 7 3 − 7( )


*45. 2 3 11 + 3 3( )

46. 3 5 2 + 5( )

47. 4 5 2 + 3 5( )

*48. 2 6 5 2 − 3 18( )

49. 3 10 5 2 − 15( )


Multiply and Simplify. You can use the FOIL method or any method for multiplying binomials.

50. 5 − 7( ) 3+ 7( )

51. 4 + 3 5( ) 2 + 5( )

*52. 2 + 3 5( ) 3− 4 5( )

53. 1− 5( ) 2 + 5( )

54. 2 2 − 5( ) 2 + 3 5( )


55. 5 2 + 6( ) 2 + 2 6( )

56. 2 − 3 5( ) 2 + 3 5( )

57. 4 7 + 3 2( ) 4 7 − 3 2( )

*58. 2 14 3 21 − 8 35( )

59. 5 6 42 − 66( )


*60. 5 + 2 7( )2

61. 5 3 − 2( )2

62. 5 3 − 2( ) 5 3 + 2( )

*63. 2 3 − 5( ) 3 + 3 5( )

*64. 3− 5( ) 3+ 5( )


65. 5 − 11( ) 5 + 11( )

66. 5 − 11( )2

67. 2 5 + 7( ) 2 5 − 7( )

68. 2 5 + 7( )2

of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra

2.4 Divide Radicals

Quotient Rule for Square Roots: ab= a

b

Use the Quotient Rule to simplify the following. The answer should not contain any square roots in the denominator. All square roots should be as simplified as possible, and all fractions should be reduced. Show all steps.

*1.

169

*2.

1825

3.

1336

4. 3 1736

5.

40x5

10x

6.

40x7

5x

*7.

12x3y5

3xy


Rationalize the denominator, and simplify. Use the quotient rule as needed. The answer should not contain any square roots in the denominator. All square roots should be as simplified as possible, and all fractions should be reduced. Show all steps. BOX answer.

*8.

12

9.

13

*11.

35

*12.

126

13. − 1015

14.

3 56


15.

53

16.

45

17.

67

*18.

512

*19.

78

*20.

245


21.

5x2

18

22.

350x

23.

23 5

24.

3 10m3

5m 6

25.

7 2015


Dividing Higher Order Roots

If an and bn are real numbers, and bn ≠ 0 , then ab

n = an

bn.

Simplify. Show all steps. Rationalize the denominator and/or use the quotient rule as needed. The answer should not contain any radicals in the denominator. All radicals should be as simplified as possible, and all fractions should be reduced. Show steps. BOX answer.

*26.

827

3

27.

64125

3

*28. − 827

3

*29.

14

3

30.

249

3

31.

25

3


*32.

363

33.

5103

*34.

32

4

35.

23

4

*36.

59

3

37.

349

3


38.

3m10

85

*39.

13y10

3

40.

1ab2c34


2.15 Conjugates & Rationalizing Denominators Simplify. Show steps. Put a BOX around answer.

*41. 5 + 7( ) 5 − 7( )

42. 4 − 7( ) 4 + 7( )

43. 7 5 − 7( )

44. 5 7( ) 7

45. 4 − 7( ) 7

46. 4 7( ) 7


Simplify. Show steps. Put a BOX around answer.

*47. 3 2 − 4( ) 3 2 + 4( )

*48. 7 − 2( ) 7 + 2( )

Fill in the blanks.

49. The conjugate of 2 + 7 is ______________________

50. The conjugate of 2 3 − 5 is ______________________

51. The conjugate of 2 3 + 5 2 is ______________________ 52. The conjugate of a + b is ______________________ Rationalize the denominator and simplify. Show all steps. There should be no square roots in the denominator. Reduce if possible. BOX answer.

*53. 1

3− 2

*54. 13 2


55. 43 −1

56. 43

57. 9

2 + 7

58. 92 7


59. 1

2 3 − 5

*60. 5

11 − 2 2

61. 6

2 3 + 5 2


*62. 8

3 2 − 2

*63. 3+ 24 − 5

64. 3+ 24 5

*65. 7 + 37 − 3


66. 3 23− 6

67. 10

10 − 6

68. 10 + 2 21

4

*67. 5 2 + 32


Below are some more problems, including some more challenging problems to do on your own paper. The challenging problems are especially appropriate for students planning to enroll in courses such as College Algebra, Business Calculus, or Precalculus.

70. 4 − 3 23 2

71. 3 2

4 + 3 2

72. 2 + 32 − 3

73. 12 − 3 2

−6

74. −1510 − 5

75. 5 6 − 320

76. 10 3 + 2 21

6

of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra

2.5 Rational Exponents Simplify. Use Laws of Exponents to simplify.

*1. b0 = *2. b−2 = Use the Laws of Exponents to complete the right side.

*3. bm ⋅bn = 4. bm( )n = Simplify. Show the intermediate step using one of the appropriate Laws from above.

*5. 512 ⋅5

12 = 6. 3

12 ⋅3

12 =

*7.

512

⎛⎝⎜

⎞⎠⎟

2

8.

1013

⎛⎝⎜

⎞⎠⎟

3

9. 1013 ⋅10

13 ⋅10

13 =

Simplify.

*10. 5 ⋅ 5 = 11. 73( )3 = 12. Fill in the blank to complete the definition below.

If n is a positive integer greater than 1 and bn is a real number, then b1n = ________


For each expression, state the base, and then circle the correct way to write it using radical notation. Then simplify the correct one.

13. −121( )12 Base: ________ Circle and simplify correct one below.

−121 − 121 12⋅121

*14. −100013 Base: ________ Circle and simplify correct one below.

−10003 − 10003

*15. 144− 12 Base: ________ Circle and simplify correct one below.

− 12⋅144 − 144 −144 −1

144

1144

16. −216( )−13 Base: ________ Circle and simplify correct one below.

1−2163

− 2163 −2163 − 13⋅216 1

2163

17. −8− 13 Base: ________ Circle and simplify correct one below.

− 83 1−83

− 183 −83


Write each expression using radical notation, and simplify if possible. If there are negative exponents, rewrite the problem using positive exponents before writing using radical notation. If the answer is not a real number, state “NOT REAL”. Show all steps. BOX answer.

*18. 1612

*19. 1013

20. 8114

21. −25( )12

*22. 8112

23. 6413 =

24. 4912

*25. −2713

26. −34313

*27. 16− 12


28. 9− 12

29. 8− 13

30. −1000− 13

31. −125( )−13

32. m12

*33. −9( )12

34. 11−13

35. n15 =


36. −2512

37. −27( )13

*38. 5x14

39. −7x13

40. 7x−13


Simplifying with Rational Exponents There are two ways to rewrite an expression with a rational exponent, where the denominator n is not zero. The second way is usually easier to compute with numbers.

*41. Follow the rule above to show two ways to rewrite bmn without using fractional

exponents—use a radical and integer exponent. Assume m and n are positive integers greater than 1 and that m/n is reduced.

bmn =

*42. Use both ways above to rewrite the expression. Then use each way to simplify 823 .

a. 823 =

b. 823

*43. Use both ways above to rewrite the expression. Then use each way to simplify 932 .

a. 932 =

b. 932 =

44. When simplifying 932 without a calculator, did you find it easier to take the square

root of 9 first and then cube it, or to cube 9 and then take the square root? Explain.


Below is a Summary of the Laws of Exponents. Assume a and b (the bases) are positive.

Note bm

bn can be written two ways. They are equivalent. So bm−n = 1

bn−m.

Often we write it the first way shown if m>n and the second way shown if n>m so that the exponent will be positive.

Also note the two ways to write amn . They are equivalent. So an( )m = amn .

bm ⋅bn = bm+n

bm( )n = bmnbm

bn= bm−n = 1

bn−m

ab( )n = anbn

ab

⎛⎝⎜

⎞⎠⎟n

= an

bn

b−n = 1bn

b0 = 1

b1n = bn

amn = an( )m = amn

ab

⎛⎝⎜

⎞⎠⎟−n

= bn

an


Write each expression using radical notation in two ways. Then, simplify one of the notations. Pick the one you find easiest to compute. Show all steps. BOX answer.

45. 2723

46. 2532

47. 8134

Write each expression using radical notation, and simplify if possible. If there are negative exponents, first rewrite the problem using positive exponents. Then rewrite using radical notation. If the answer is not a real number, state NOT REAL. Show all steps. BOX answer.

*48. −1634

49. −2532

50. −25( )32


*51. 8− 23

52. 4− 52

*53. −27( )43

54. −32( )25

*55. 49

⎛⎝⎜

⎞⎠⎟

32

56. − 278

⎛⎝⎜

⎞⎠⎟

23


*57. − 452

58. −952

*59. −36( )12

60. 4925

⎛⎝⎜

⎞⎠⎟− 32

61. − 278

⎛⎝⎜

⎞⎠⎟− 23


*62. 2x( )35

63. 3x( )54

*64. 2x35

65. 3x54

*66. −27( )−23

67. − 8125

⎛⎝⎜

⎞⎠⎟− 23

*68. x− 16

69. m− 25


*70. 4

5x− 12

*71. − 4− 32

72. 2x − 3( )23

Use the Laws of Exponents to write each expression with a base of x—simply one base of x raised to some power. After you do that, if the exponent is negative, rewrite with a positive exponent, and if the exponent is fractional, write in radical form. Show steps. Box answer.

*73. x34 ⋅ x

− 74

74. x415 ⋅ x

715

*75.

x34

x16

76.

m710

m110


Advanced Rational Exponents Below are some challenging problems you might try. These are appropriate for students planning to enroll in courses such as College Algebra, Business Calculus, or Precalculus. Assume all variables are positive. Use the laws of exponents to simplify each expression. Do not write any answers with negative exponents. Show all steps. BOX Answer.

*77. 27u3( )23 *78.

x14 x

− 12

x23

79. x16x

− 56

x13

*80. a−2b3( )

18

a−3b( )−14

81. mb m−1b3( )0

m−2b4( )−12

82. 8u3

125m9

⎛⎝⎜

⎞⎠⎟

−23

*83. a318 84. a525

*85. 364 86. x1015

*87. x + 4( )48 Rewrite using rational exponents. Then write so the base is raised to a single power. Then write in radical form. Assume all variables are positive. Show all steps. BOX answer.

*88. y23 ⋅ y6 *89. x5

x6

90. n23 nn4

*91. 23 5

*92. x34 y 23

of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra

2.6 Radical Equations POWER RULE FOR EQUATIONS: If both sides of an equation are raised to the same power, all the solutions of the original equation are also among the solutions of the new equation. The Power Rule does NOT state that both equations necessarily have the same solution. Any time you raise both sides of an equation to the same power, the new equation might have more solutions than the original equation. This is an extremely important point, and why it’s especially crucial that if you use the power rule to solve an equation that you always check any solutions in the ORIGINAL equation. To solve an equation containing one or more radicals, we use the Power Rule for Equations. How to Solve an Equation containing one or more Radicals Step 1: Isolate one radical on one side of the equation. Step 2: Use the Power Rule to raise each side of the equation to the power of the index of

the radical—if it’s a square root, square both sides; if it’s a cube root, cube both sides, etc.

Step 3: Simplify both sides of the equations. Cautionary Note: If one side contains more than

one term, and you are squaring that side, you do not simply square each term. You might be squaring a binomial on that side, in which case, use the FOIL method.

Step 4: If after simplifying each side, the equation has another radical, repeat steps 1-‐3. Step 5: Once there are no more radicals, solve the equation. Step 6: Check each solution in the ORIGINAL equation. The last step is crucial and very important when using the Power Rule. Always Check! If a proposed solution does not check in the original equation, we call it an extraneous root. It is NOT a solution unless it checks in the original.


Solve each equation, and check each solution. Write solutions that check on the blank space provided. Show all steps, including steps to checking each solution.

*1. x = 4 Solution(s): __________________ Show check(s).

2. 2 m = 14 Solution(s): __________________ Show check(s).

*3. x + 4 = 5 Solution(s): __________________ Show check(s).

4. 3− 2x = 2 Solution(s): __________________ Show check(s).

*5. x = −3 Solution(s): __________________ Show check(s).


6. x + 9 = 2 Solution(s): __________________ Show check(s).

*7. x − 3 − 2 = 4 Solution(s): __________________ Show check(s).

8. x +1 − 3 = 2 Solution(s): __________________ Show check(s).

*9. 2x − 3 + 3 = 6 Solution(s): __________________ Show check(s).


10. 2x +1 +1= 4 Solution(s): __________________ Show check(s).

*11. 5x − 9 = 2x − 3 Solution(s): __________________ Show check(s).

12. 2x − 73 = 8 − 3x3 Solution(s): __________________ Show check(s).

*13. x + 5 = x +1 Solution(s): __________________ Show check(s).


14. x − 8 = x − 2 Solution(s): __________________ Show check(s).

*15. x + x + 5 = 7 Solution(s): __________________ Show check(s).

16. 2 x + 3 +1= x + 4 Solution(s): __________________ Show check(s).


*17. 2x3 + 5 = 9 Solution(s): __________________ Show check(s).

18. 3 x3 = x2 +17x3 Solution(s): __________________ Show check(s).

*19. x − 43 − 5 = −7 Solution(s): __________________ Show check(s).

20. 2x + 33 + 3 = 5 Solution(s): __________________ Show check(s).


*21. 7x − 5 = 5 − 7x Solution(s): __________________ Show check(s).

22. x + 84 = 2x4 Solution(s): __________________ Show check(s).

*23. x + 34 − 5 = −3 Solution(s): __________________ Show check(s).

24. 2x + 34 + 9 = 12 Solution(s): __________________ Show check(s).


*25. x − x + 9 = 1 Solution(s): __________________ Show check(s).

26. x − 7 + x = 7 Solution(s): __________________ Show check(s).

*27. 2x −14 − x = −1 Solution(s): __________________ Show check(s).


28. x − 8 = x − 2 Solution(s): __________________ Show check(s).

*29. x − 4 + x = 6 Solution(s): __________________ Show check(s).

*30. x + 2 + 4 = x Solution(s): __________________ Show check(s).


31. 26 −11x = 4 − x Solution(s): __________________ Show check(s).

*32. −3x +16 +1= −4x + 25 Solution(s): __________________ Show check(s).

of 98 Unit 2 Harland 2.7 Pythagorean Theorem & Distance Formula Intermediate Algebra

2.7 Pythagorean Theorem & Distance Formula The Pythagorean Theorem is a very famous mathematical theorem. In the diagram on the left below, look at the triangle in the middle. One leg has length a, and the other leg has length b, and the hypotenuse has length c. The square with side of a has an area of a2, the square with side of b has an area of b2, and the square with side of c has an area of c2. The Pythagorean Theorem states that the sum of the areas of the two smaller squares is the same area as the larger square. The Pythagorean Theorem is often stated this way: The square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (called legs). If we are looking at a triangle where the lengths of the legs are a and b, and the length of the hypotenuse is c, we often write this as shown on the diagram on the right.

There are several proofs showing why the Pythagorean Theorem is true. I’ve include one proof on a video. You can check find other proofs by searching the internet. 1. Draw a right triangle with legs of lengths x and y, and hypotenuse of length z. Then

use the Pythagorean Theorem to state the equation that is true for this triangle using x, y, and z.


For each right triangle having the sides indicated, draw a picture of the triangle, and use the Pythagorean Theorem to find the missing side. Write the exact answer (simplifying any radicals if possible), and then use a calculator to round the answer to one place after the decimal point. Show work. *2. One leg is 5” and hypotenuse is 8”. Exact length of other leg:_____________ Rounded length of other leg:_____________ 3. One leg is 7’ and hypotenuse is 15’. Exact length of other leg:_____________ Rounded length of other leg:_____________


*4. The legs are 3 m and 5 m Exact length of hypotenuse:_____________ Rounded length of hypotenuse:_____________ 5. The legs are 7 cm and 15 cm Exact length of hypotenuse:_____________ Rounded length of hypotenuse:_____________ *6. One leg is 8 cm; hypotenuse is 12 cm. Exact length of other leg:_____________ Rounded length of other leg:_____________


*7. The legs are 7 m and 24 m Exact length of hypotenuse:_____________ Rounded length of hypotenuse:_____________ *8. One leg is2 2 ; hypotenuse is 8 Exact length of other leg:_____________ Rounded length of other leg:_____________

9. The legs are 5 and 2 3 Exact length of hypotenuse:_____________ Rounded length of hypotenuse:_____________


DISTANCE FORMULA If (x1.y1) and (x2.y2) are two points, the distance, d, between the two points, is shown below. This can be shown to be true by using the Pythagorean Theorem. An picture of how to derive the Distance Formula is below.

*10. State the distance formula between points x1, y1( ) and x2, y2( ) . 11. State the distance formula between points a,b( ) and c,d( ) .


Plot the 2 points and draw a line segment between them and call it d. Then compute the exact distance, d, between the points, simplifying any radical if possible. Then use a calculator to round to the nearest tenth—one place after decimal point. Show all steps. *12. −3,5( ) and 6,−2( ) Exact: ___________ Rounded: ____________

*13. −3,−6( ) and 4,−8( ) Exact: ___________ Rounded: ____________

14. −1,3( ) and 2,4( ) Exact: ___________ Rounded: ____________

x

y

5

5

- 5

- 5

x

y

5

5

- 5

- 5

x

y

5

5

- 5

- 5


*15. −6,−5( ) and −8,−8( ) Exact: ___________ Rounded: ____________

*16. 0,4( ) and −9,3( ) Exact: ___________ Rounded: ____________

*17. 12,−3⎛

⎝⎜⎞⎠⎟ and 4

12,6⎛

⎝⎜⎞⎠⎟ Exact: ___________ Rounded: ____________

x

y

5

5

- 5

- 5

x

y

5

5

- 5

- 5

x

y

5

5

- 5

- 5


*18. −2,4( ) and −2,−6( ) Exact: ___________ Rounded: ____________

Use the distance formula to find the exact distance between each pair of points. Simplify any radical if possible. Then use a calculator to round to the nearest tenth—one place after decimal point. Show all steps. 19. 5,−5( ) and −7,6( ) Exact: ___________ Rounded: ____________ 20. −4,−9( ) and −2,−3( ) Exact: ___________ Rounded: ____________

21. 12,−1⎛

⎝⎜⎞⎠⎟ and − 1

2,7⎛

⎝⎜⎞⎠⎟ Exact: ___________ Rounded: ____________

x

y

5

5

- 5

- 5

of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra

2.8 Complex Numbers Imaginary Number and Powers of i *1. Define the imaginary unit, i : i = _________ *2. Evaluate: i2 = ________ *3. Is i a real number?

*4. Is −9 a real number?

*5. Is −9 in its most simplified form?

*6. Show the steps to simplify: −9

*7. If −9 is not a real number, what kind of number is it?

*8. Complete the statement: If b > 0 , then −b = ___________ It is NEVER simplified to leave a negative number under a square root symbol. Anytime there is a negative number under a square root, the first step is to rewrite it as the square root of the absolute value of that number (so it’s positive) times the square root of -‐1—which is i . The i is NOT under the square root symbol. To clarify this, some people write the i in front of the square root if the square root does not simplify to a rational number. See below for an example:

−18 = 18 ⋅ −1 = 3 2i Note the i is NOT under the square root sign. Be careful NOT to write 3 2i Or you may do it this way:

−18 = −1 ⋅ 18 = i ⋅3 2 = 3i 2

So you may see −18 written either of these ways in simplified form: 3i 2 or 3 2i You need to get used to seeing it in either form. It is easier to pick out the imaginary part if the i is at the end, but make sure you do not extend the square root symbol over the i .


Simplify. There should be no negative numbers under the square root, and all radicals need to be simplified. Show all steps.

*9. −25

*10. −18

11. −90

12. −28 Any number that can be written in the the form bi where b is a real number and i = −1 is called an imaginary number (or pure imaginary number). Any number that can be written in the form a + bi where a and b are real numbers and i = −1 is an imaginary number is called a complex number. a is called the real part and b is called the imaginary part of a + bi . Note that the imaginary part is only the real number b , not bi . 13. Below are examples of numbers that can be put in the form a + bi so it is easy to

identify the real and imaginary parts. See the examples at the beginning and then fill in the real and imaginary parts for the rest. Simplify a and b as needed when you put it in the form a + bi so that the real and imaginary parts are simplified.

Number In form a + bi Real part Imaginary Part 3− 5i 3− 5i 3 −5 2 3 2 3 + 0i 2 3 0 8i 0 + 8i 0 8 −7 + i −7 + i or −7 +1i −7 1 −3+ 13i −3+ 13i

13 − i

−2 5i

17

−36

5 − −16

75 + −13


Simplify each of the following. Answer to each will either be 1, −1 , i , or −i . Show your reasoning and steps. 14. i2 *15. i3

*16. i4 *17. i5

*18. i6 *19. i7

20. i8 *21. i20

*22. i32 23. i40

*24. i22 *25. i33

*26. i103 27. i17

*28. i57 *29. i30

30. i102 *31. i23

*32. i44 33. i501

*34. i50 35. i69

These next two are a little tricky. You can do them!

*36. i−1 37. i−2


Below are some important definitions and properties. It’s very important to note the property below is only true if both a and b are positive.

If a ≥ 0 and b ≥ 0 , then a ⋅ b = ab For any numbers a and b, positive or negative, the following are always true.

ab= a

b a a = a a( )2 = a Definition of i (Note that i is NOT a variable) : i = −1 and i2 = −1

If b > 0 , then −b = bi or −b = i b If there is a negative number under a square root, use the above property for the first step. If a negative number is under a square root, the FIRST STEP is to rewrite the expression using i so only a positive number is under the square root. Then, if possible, simplify the square root using properties of square roots.

Example: −6 = 6i NOTE: i is NOT under the square root symbol. You can also write it this way: −6 = i 6 Simplify. If a negative number is under a square root, the FIRST STEP is to rewrite the expression using i so only a positive number is under the square root. Then, if possible, simplify the square root using properties of square roots. Show steps. BOX answer.

*38. −50

39. −15

40. −90

*41. −6 −9


Simplify. If a negative number is under a square root, the FIRST STEP is to rewrite the expression using i so only a positive number is under the square root. Then, if possible, simplify the square root using properties of square roots. Show steps. BOX answer.

*42. 5 −45

43. 2 −54

For any number, a a = a and a( )2 = a 44. In the following few examples, you may use the properties above. These are special

cases where you do not have to first rewrite it with i since the properties are true for all values of a. You will get the same answer using the properties above, or by first rewriting each square root with i and a positive number under the square root before simplifying.

*a. −11 ⋅ −11

*b. −41 ⋅ −41

c. −73 ⋅ −73

d. −26( )2 This property applies only when both a and b are positive: a ≥ 0 and b ≥ 0 , then a ⋅ b = ab Simplify. If a negative number is under a square root, the FIRST STEP is to rewrite the expression using i so only a positive number is under the square root. Then, if possible, simplify the square root using properties of square roots. Show steps. BOX answer.

*45. −2 ⋅ −8

46. −27 ⋅ −3


47. −5 ⋅ −6

*48. 2 −12

49. −5 −63

*50. −6 ⋅ −10

51. −10 ⋅ −15

*52. 3 −7 ⋅2 −14

53. −5 −21 ⋅2 −15


Product Rule: If a ≥ 0 and b ≥ 0 , then a ⋅ b = ab

Quotient Rule: ab= a

b

*54. −182

55. −18−2

*56. −40−5

57. −120−15

58. −5 33 ⋅4 −22


Add complex numbers like you add like terms even though i is not a variable. Add the real numbers together, and add the imaginary numbers together. Simplify the real and

imaginary parts as needed.

To subtract, distribute the negative sign: a + bi( )− c + di( ) = (a − c)+ (b − d)i Add or Subtract as indicated. Simplify the real and imaginary parts. Write your answer as a real number, a pure imaginary number, or as a complex number in the form a + bi or a − bi . Show steps. BOX answer. *59. 3− 2i( ) + 6 + 4i( ) 60. −3+ 7i( ) + −4 − 7i( ) 61. 3+ 2i( ) + −3− 5i( ) 62. 3+ 2i( )− −3+ i( )

63. 6 3 + 2i( ) + −3 3 + 5 2i( )

64. 6 3 + 2i( )− 3 3 − 5 2i( )

65. 3− −9( )− 2 + −16( )

66. 5 3 − −8( ) + 12 + −2( )

a + bi( ) + c + di( ) = (a + c)+ (b + d)i


Multiply. Simplify the real and imaginary parts. Do not leave i2 in answer as that can be simplified to -‐1. Write your answer as a real number, a pure imaginary number, or as a complex number in the form a + bi or a − bi . Show all steps. BOX answer. *67. −4i ⋅3i *68. −5 ⋅−2i 69. −i ⋅−i 70. −4i ⋅9

71. −9 ⋅2i 72. −7 9 − 7i( ) *73. 3i 5 + 7i( ) 74. −2i 4 + 3i( )

75. −6 4 + −6( ) *76. 2 + 3i( ) 6 − 5i( ) 77. −1− 2i( ) −1− i( )


Remember: a + b( )2 = a + b( ) a + b( )

*78. 7 + i( )2 *79. 7 + i( ) 7 − i( )

*80. 2 + 3i( )2 81. 2 + 3i( ) 2 − 3i( )

82. 8 − i( )2


*83. 4 + 3i( ) 4 − 3i( ) 84. −5 − 2i( ) −5 + 2i( ) *85. 3+ 5i( ) 3− 5i( )

86. 3− 5i( )2 *87. 6 + 5i( ) 6 − 5i( )


a + bi and a − bi are called Complex Conjugates. Multiply and simplify, showing all steps. You should get different answers for these. 88. a + b( ) a − b( ) 89. a + bi( ) a − bi( ) Simplify. Show all steps.

90. 2 3 − 3i( ) 2 3 + 3i( )

91. 5 − 3( ) 5 + 3( )

92. 2 3 + −7( ) 2 3 − −7( )

93. 2 3 + 7( ) 2 3 − 7( )


Simplify. Do not leave any power of i in answer as any power of i can be simplified to i , −i , −1, or 1 . Write your answer as a real number, a pure imaginary number, or as a complex number in the form a + bi or a − bi . Show all steps. BOX answer. These are a bit more challenging.

94. −5 ⋅ −2 ⋅ −3

95. −5 ⋅ −3 ⋅ −2 ⋅ −1 96. 3i ⋅2i2 ⋅5i3

97. 3i3 2i2 − 5i( )

98. 2i − 5i3( ) 2i + 5i3( )

99. 2i2 − 5i6( )2

of 98 Unit 2 Harland 2.9 Dividing Complex Numbers Intermediate Algebra

2.9 Dividing Complex Numbers Divide. Simplify. Rationalize all denominators, which means there should not be any imaginary numbers or radicals in the denominator. All radicals should be simplified, with no negative numbers in the radical, and all fractions should be reduced. Do not leave i2 in answer as that can be simplified to -‐1. Write answer as a real number, a pure imaginary number, or as a complex number in the form a + bi or a − bi . Show all steps. BOX answer.

*1. 3i

2. − 5i

3. 32

*4. −52i

5. 2i3

6. 7

5 3i


7. 2i−6

*8. 9 − 6i3i

*9. 9 − 6i3

10. 4 −15i−3i

*11. 34i

12. 20 − 5i10i


*13. −8 +12i6i

14. 4 − 3i6i

15. 20 − 5i10

16. 10 − 2i−10


Complex Conjugates a + bi and a − bi are called complex conjugates. Below is the product of two complex conjugates. a + bi( ) a − bi( ) = a2 − b2i2 = a2 − b2 (−1) = a2 + b2 This essentially is first the difference of two squares, but since there was an i2 , −1 is multiplied by b2 to end up with a2 + b2 for the simplified answer. Remember that a and b represent the real and imaginary parts. b is the coefficient of i. The product, a2 + b2 , is a real number, since there is no i in the answer. This will be used to rationalize denominator that are complex numbers. If both the real and imaginary parts are rational numbers and/or square roots, the product of complex conjugates will not only be real, it will also rational. Note that a2 + b2 is a SUM of two positive numbers since if both a and b are real numbers, then when you square each of them, you get positive numbers. So any time you multiply conjugates, you end up adding two positive numbers. Example: State the conjugate of 5 − 2i . Then multiply 5 − 2i by its conjugate. The conjugate of 5 − 2i is 5 + 2i . If we multiply 5 − 2i by its conjugate, we will get a rational number in the denominator. 5 − 2i( ) 5 + 2i( ) = 25 − 4i2 = 25 − 4(−1) = 25 + 4 = 29 So when we multiply 5 − 2i by its conjugate, we get 29, a rational number. Note that since a + bi( ) a − bi( ) == a2 + b2 , the quick way to multiply two conjugates is to add the squares of the real part and the imaginary part (the coefficient of i). It’s only a sum of two squares if they are complex conjugates! 5 − 2i( ) 5 + 2i( ) == 52 + 22 = 25 + 4 = 29 Be really careful anytime the imaginary part is 1 or -‐1. You won’t usually see the coefficient of 1 or -‐1. Remember that b, the imaginary part is the coefficient of i. Example: State the real part, imaginary part, and conjugate of 2 − i . Then multiply 2 − i its conjugate and simplify. Solution: a = 2 b = −1 Conjugate: 2 + i 2 − i( ) 2 + i( ) = 22 +12 = 4 +1= 5


State the real and imaginary part of each complex number. Then state its conjugate. Then multiply each complex number by its conjugate and simplify. Show all steps. 17. 4 + 2i a = ________ b = ________ Conjugate:___________________

Multiply the complex number above by its conjugate and simplify. Show all steps.

18. 7 − 3i a = ________ b = ________ Conjugate:___________________


19. 5 + 2i a = ________ b = ________ Conjugate:___________________

Multiply the complex number above by its conjugate and simplify. Show all steps. 20. 6 + i a = ________ b = ________ Conjugate:___________________


21. 2 6 − 3 2i a = ________ b = ________ Conjugate:___________________ Rationalize the denominator and simplify. Show all steps. Write answer in form a + bi .

*22. 53+ i


23. 3

5 − 2i

*24. 103− 4i

25. −2

−1+ 4i

*26. 4i5 + i


27. −22i3− 2i

*28. 3+ 2i7 + 2i

29. 5 − i2 − i

30. 3 + 5i5 + 3i


Mixed Practice with Complex Numbers

Simplify as much as possible. There should be no negative numbers in square roots, and no irrational or imaginary numbers in the denominator. Write answers as a real number, a pure imaginary number or as a complex number in the form a + bi . Show all steps. BOX answer.

31. −3 −20 + 5i −18 32. 5 −3 ⋅2 −6

33. 45−8

34. 8 − 2i( )− 6 − 3 2i( )

35. 4 − 5i( )2 36. 3 2 − 4i( ) 3 2 + 4i( )

37. i39 38. 3

6 + 2i

of 98 Unit 2 Harland 2.10 Review Exercises Intermediate Algebra

2.10 Review Exercises

Answer all problems completely. Show all steps and work. It’s important to be able to do all problems on your own in any order without referring to examples or notes.

Simplify. Assume all variables are positive. All radicals need to be simplified. All denominators needs to be rational, and all fractions need to be reduced.

1. 14x15 ⋅ 21x7

2. −5mn3 108m16n13

3. −3xy −40x11y153

4. 3x 125 − 5 50x2

5. 5x2 x54 − x3 16x4



6. 5 10 3 2 − 15( )

7. 2 3 + 3 5( ) 3 3 − 4 5( )

8. 2 3 + 3 5( )2

9. 2 3 + 3 5( ) 2 3 − 3 5( )

10. − 252 15



11. −3549x3

12. −12

3 2 + 2

Write each expression using radical notation, and simplify if possible. If there are negative exponents, rewrite the problem using positive exponents before writing in radical notation. If the answer is not a real number, state that. Show all steps.

13. −843

14. 81− 34


Solve each equation, and check each solution. Show all steps, including all steps to checking each solution. PUT a BOX around solution(s) that check.

15. 2x −1 + 2 = 5

16. x +1= 8 − x + 5

17. 3x +1 − x + 4 = 1


Find the exact answer (simplifying any radicals if possible) of the missing side of the right triangle, where a and b represent the legs and c represents the hypotenuse. and then use a calculator to round the answer to one place after the decimal point. Draw a picture of the triangle and Show work. 18. a = 5; b = 11; c = ______ 19. a = 9; c = 13; b = ______ 20. Use the distance formula to compute the exact distance between −1,1( ) and

−3,−7( ) , simplifying any radical if possible. Write the exact answer. Then use a calculator to round to the nearest tenth—one place after decimal point. Show all steps.

Exact answer: __________ Approximation: ____________ Simplify. There should be no powers of i, no negative numbers under a square root, and no square roots or imaginary numbers in the denominator. All radicals and fractions need to be simplified. Show all steps. 21. i43

22. 3 + 2i( )− 5 3 − 4 2i( )

23. −7 ⋅3 −35


Simplify. There should be no powers of i, no negative numbers under a square root, and no square roots or imaginary numbers in the denominator. All radicals and fractions need to be simplified. Show all steps.

24. 4 2 + 3i( ) 4 2 − 3i( )

25. 7 − 3i( )2

26. 3− −15

−6i

27. 2i

4 − 2i

intermediate’algebra’unit’2’ radicals,’rational’exponents ... · page 5 of...

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