1. Rational Exponents, Radicals, and Complex Numbers
CHAPTER 10
Rational Exponents, Radicals, and Complex Numbers
10.1 Radical Expressions and Functions
10.2 Rational Exponents
10.3 Multiplying, Dividing, and Simplifying Radicals
10.4 Adding, Subtracting, and Multiplying Radical Expressions
10.5 Rationalizing Numerators and Denominators of Radical Expressions
10.6 Radical Equations and Problem Solving
10.7 Complex Numbers
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2. Radical Expressions and Functions
10.1
Radical Expressions and Functions
1. Find the nth root of a number.
2. Approximate roots using a calculator.
3. Simplify radical expressions.
4. Evaluate radical functions.
5. Find the domain of radical functions.
6. Solve applications involving radical functions.
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3. Copyright © 2011 Pearson Education, Inc.
nth root: The number b is an nth root of a number a if bn = a.
Evaluating nth roots
When evaluating a radical expression , the sign of a and the index n will determine possible outcomes.
If a is nonnegative, then , where and bn = a.
If a is negative and n is even, then there is no real-number root.
If a is negative and n is odd, then , where b is negative and bn = a.
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4. Copyright © 2011 Pearson Education, Inc.
Example 1
Evaluate each root, if possible. a. b. c.
Solution
Solution
is not a real number because there is no real number whose square is –100.
Solution
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5. Copyright © 2011 Pearson Education, Inc.
continued
Evaluate each root, if possible. d. e. f.
Solution
Solution
Solution
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6. Copyright © 2011 Pearson Education, Inc.
continued
Evaluate each root, if possible. g. h.
Solution
Solution
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7. Copyright © 2011 Pearson Education, Inc.
Some roots, like are called irrational because we cannot express their exact value using rational numbers. In fact, writing with the radical sign is the only way we can express its exact value. However, we can approximate using rational numbers.
Approximating to two decimal places:
Approximating to three decimal places:
Note: Remember that the symbol, , means “approximately equal to.”
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8. Copyright © 2011 Pearson Education, Inc.
Example 2
Approximate the roots using a calculator or table in the endpapers. Round to three decimal places. a. b. c.
Solution
Solution
Solution
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9. Copyright © 2011 Pearson Education, Inc.
Example 3
Find the root. Assume variables represent nonnegative values. b. c. d.
Solution
Because (y2)2 = y4.
Solution
Because (6m3)2 = 36m6.
Solution
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10. Copyright © 2011 Pearson Education, Inc.
continued
Find the root. Assume variables represent nonnegative values. e. f.
Solution
Solution
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11. Copyright © 2011 Pearson Education, Inc.
Example 4
Find the root. Assume variables represent any real number. a. b. c.
Solution
Solution
Solution
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12. Copyright © 2011 Pearson Education, Inc.
continued
Find the root. Assume variables represent any real number. d. e. c.
Solution
Solution
Solution
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13. To find f(3), substitute 3 for x and simplify.
Radical function: A function containing a radical expression whose radicand has a variable.
Example 5a
Given f(x) = find f(3).
Solution
To find f(3), substitute 3 for x and simplify.
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14. Copyright © 2011 Pearson Education, Inc.
Example 6
Find the domain of each of the following. a. b.
Solution Since the index is even, the radicand must be nonnegative.
Domain:
Solution The radicand must be nonnegative.
Domain:
Conclusion The domain of a radical function with an even index must contain values that keep its radicand nonnegative.
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15. Copyright © 2011 Pearson Education, Inc.
Example 7
If you drop an object, the time (t) it takes in seconds
to fall d feet is given by Find the time it
takes for an object to fall 800 feet.
Understand We are to find the time it takes for an object to fall 800 feet.
Plan Use the formula , replacing d with 800.
Execute
Replace d with 800.
Divide within the radical.
Evaluate the square root.
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continued
Answer It takes an object seconds to fall feet.
Check We can verify the calculations, which we will leave to the viewer.
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17. Copyright © 2011 Pearson Education, Inc.
For which square root is –12.37 the approximation for? a) b) c) d)
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18. Copyright © 2011 Pearson Education, Inc.
For which square root is –12.37 the approximation for? a) b) c) d)
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19. Copyright © 2011 Pearson Education, Inc.
Evaluate.
a) 0.2
b) 0.02 c) d)
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20. Copyright © 2011 Pearson Education, Inc.
Evaluate.
a) 0.2
b) 0.02 c) d)
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21. Copyright © 2011 Pearson Education, Inc.
Find the domain of f(x) = a) b) c) d)
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22. Copyright © 2011 Pearson Education, Inc.
Find the domain of f(x) = a) b) c) d)
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23. Copyright © 2011 Pearson Education, Inc.
10.2
Rational Exponents
1. Evaluate rational exponents.
2. Write radicals as expressions raised to rational exponents.
3. Simplify expressions with rational number exponents using the rules of exponents.
4. Use rational exponents to simplify radical expressions.
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24. Rational exponent: An exponent that is a rational number.
Rational Exponents with a Numerator of 1
a1/n = where n is a natural number other than 1.
Note: If a is negative and n is odd, then the root is negative.
If a is negative and n is even, then there is no real number root.
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25. Copyright © 2011 Pearson Education, Inc.
Example 1
Rewrite using radicals, then simplify if possible. a. 491/2 b. 6251/4 c. (216)1/3
Solution a. b. c.
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26. Copyright © 2011 Pearson Education, Inc.
continued
Rewrite using radicals, then simplify. d. (16)1/ e. 491/2 f. y1/6
Solution d. e. f.
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27. Copyright © 2011 Pearson Education, Inc.
continued
Rewrite using radicals, then simplify. g. (100x8)1/ h. 9y1/5 i.
Solution d. e. f.
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28. Copyright © 2011 Pearson Education, Inc.
General Rule for Rational Exponents
where a  0 and m and n are natural numbers other than 1.
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29. Copyright © 2011 Pearson Education, Inc.
Example 2
Rewrite using radicals, then simplify, if possible.
a. 272/3 b. 2433/5 c. 95/2
Solution a. b. c.
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30. Copyright © 2011 Pearson Education, Inc.
continued
Rewrite using radicals, then simplify, if possible.
d. e. f.
Solution d. e. f.
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31. Copyright © 2011 Pearson Education, Inc.
Negative Rational Exponents
where a  0, and m and n are natural numbers with n  1.
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32. Copyright © 2011 Pearson Education, Inc.
Example 3
Rewrite using radicals; then simplify if possible.
a. 251/2 b. 272/3
Solution a. b.
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33. Copyright © 2011 Pearson Education, Inc.
continued
Rewrite using radicals; then simplify if possible.
c. d.
Solution c.
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34. Copyright © 2011 Pearson Education, Inc.
Example 4
Write each of the following in exponential form. a. b.
Solution a. b.
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35. Copyright © 2011 Pearson Education, Inc.
continued
Write each of the following in exponential form. c. d.
Solution c. d.
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36. Copyright © 2011 Pearson Education, Inc.
Rules of Exponents Summary
(Assume that no denominators are 0, that a and b are real numbers, and that m and n are integers.)
Zero as an exponent: a0 = 1, where a 0.
00 is indeterminate.
Negative exponents:
Product rule for exponents:
Quotient rule for exponents:
Raising a power to a power:
Raising a product to a power:
Raising a quotient to a power:
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37. Copyright © 2011 Pearson Education, Inc.
Example 5a
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Use the product rule for exponents. (Add the exponents.)
Add the exponents.
Simplify the rational exponent.
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38. Copyright © 2011 Pearson Education, Inc.
Example 5b
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Use the product rule for exponents. (Add the exponents.)
Rewrite the exponents with a common denominator of 6.
Add the exponents.
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39. Copyright © 2011 Pearson Education, Inc.
Example 5c
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Use the quotient for exponents. (Subtract the exponents.)
Rewrite the subtraction as addition.
Add the exponents.
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40. Copyright © 2011 Pearson Education, Inc.
Example 5d
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Add the exponents.
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41. Copyright © 2011 Pearson Education, Inc.
Example 5e
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
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42. Copyright © 2011 Pearson Education, Inc.
Example 5f
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
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43. Copyright © 2011 Pearson Education, Inc.
Example 5g
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
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44. Copyright © 2011 Pearson Education, Inc.
Example 6
Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values.
a b.
Solution
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45. Copyright © 2011 Pearson Education, Inc.
continued
Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values. c.
Solution
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46. Copyright © 2011 Pearson Education, Inc.
Example 7
Perform the indicated operations. Write the result using a radical. a. b.
Solution a. b.
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47. Copyright © 2011 Pearson Education, Inc.
continued
Perform the indicated operations. Write the result using a radical. c.
Solution c.
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48. Copyright © 2011 Pearson Education, Inc.
Example 8
Write the expression below as a single radical. Assume that all variables represent nonnegative values.
Solution
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49. Copyright © 2011 Pearson Education, Inc.
Simplify. a) b) c) d)
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50. Copyright © 2011 Pearson Education, Inc.
Simplify. a) b) c) d)
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51. Copyright © 2011 Pearson Education, Inc.
Simplify.
a) 5
b) 25
c) 25
d) 5
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52. Copyright © 2011 Pearson Education, Inc.
Simplify.
a) 5
b) 25
c) 25
d) 5
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53. Copyright © 2011 Pearson Education, Inc.
Simplify.
a) 4 b)
c) 4 d)
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54. Copyright © 2011 Pearson Education, Inc.
Simplify.
a) 4 b)
c) 4 d)
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55. Multiplying, Dividing, and Simplifying Radicals
10.3
Multiplying, Dividing, and Simplifying Radicals
1. Multiply radical expressions.
2. Divide radical expressions.
3. Use the product rule to simplify radical expressions.
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56. Copyright © 2011 Pearson Education, Inc.
Product Rule for Radicals
If both and are real numbers, then
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57. Copyright © 2011 Pearson Education, Inc.
Example 1
Find the product and simplify. Assume all variables represent positive values.
a. b.
Solution
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58. Copyright © 2011 Pearson Education, Inc.
continued
Find the product and simplify. Assume all variables represent positive values.
c. d.
Solution
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59. Copyright © 2011 Pearson Education, Inc.
continued
Find the product and simplify. Assume all variables represent positive values.
e. f.
Solution
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60. Copyright © 2011 Pearson Education, Inc.
continued
Find the product and simplify. Assume all variables represent positive values. g.
Solution
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61. Copyright © 2011 Pearson Education, Inc.
Raising an nth Root to the nth Power
For any nonnegative real number a,
Quotient Rule for Radicals
If both and are real numbers, then
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62. Copyright © 2011 Pearson Education, Inc.
Example 2
Simplify. Assume variables represent positive values. a.
Solution b. c. a. c. b.
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63. Copyright © 2011 Pearson Education, Inc.
continued
Simplify. Assume variables represent positive values. d.
Solution e. d. e.
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64. Copyright © 2011 Pearson Education, Inc.
Simplifying nth Roots
To simplify an nth root,
1. Write the radicand as a product of the greatest possible perfect nth power and a number or an expression that has no perfect nth power factors.
2. Use the product rule when a is the perfect nth power.
3. Find the nth root of the perfect nth power radicand.
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65. Copyright © 2011 Pearson Education, Inc.
Example 3
Simplify.
a. b.
Solution
Solution
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66. Copyright © 2011 Pearson Education, Inc.
continued
Simplify.
c. d.
Solution
Solution
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67. Copyright © 2011 Pearson Education, Inc.
Example 4a
Simplify the radical using prime factorization.
Solution
Write 686 as a product of its prime factors.
The square root of the pair of 7s is 7.
Multiply the prime factors in the radicand.
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68. Copyright © 2011 Pearson Education, Inc.
continued
Simplify the radical using prime factorization.
b. c. b.
Solution c.
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69. Copyright © 2011 Pearson Education, Inc.
Example 5a
Simplify.
Solution
The greatest perfect square factor of 32x5 is 16x4.
Use the product rule of square roots to separate the factors into two radicals.
Find the square root of 16x4 and leave 2x in the radical.
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70. Copyright © 2011 Pearson Education, Inc.
Example 5b
Simplify
Solution
The greatest perfect square factor of 96a4b is 16a4.
Use the product rule of square roots to separate the factors into two radicals.
Find the square root of 16a4 and leave 6b in the radical.
Multiply 2 and 4.
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71. Copyright © 2011 Pearson Education, Inc.
continued
Simplify.
c. d.
Solution
Solution
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72. Copyright © 2011 Pearson Education, Inc.
Example 6
Find the product or quotient and simplify the results. Assume that variables represent positive values.
a b.
Solution
Solution
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73. Copyright © 2011 Pearson Education, Inc.
continued
Find the product or quotient and simplify the results. Assume that variables represent positive values.
c d.
Solution
Solution
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74. Copyright © 2011 Pearson Education, Inc.
Simplify. Assume all variables represent nonnegative numbers. a) b) c) d)
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75. Copyright © 2011 Pearson Education, Inc.
Simplify. Assume all variables represent nonnegative numbers. a) b) c) d)
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76. Copyright © 2011 Pearson Education, Inc.
Simplify. a) b) c) d)
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77. Copyright © 2011 Pearson Education, Inc.
Simplify. a) b) c) d)
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78. Adding, Subtracting, and Multiplying Radical Expressions
10.4
Adding, Subtracting, and Multiplying Radical Expressions
1. Add or subtract like radicals.
2. Use the distributive property in expressions containing radicals.
3. Simplify radical expressions that contain mixed operations.
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79. Copyright © 2011 Pearson Education, Inc.
Like radicals: Radical expressions with identical radicands and identical root indices.
Adding Like Radicals
To add or subtract like radicals, add or subtract the coefficients and leave the radical parts the same.
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80. Copyright © 2011 Pearson Education, Inc.
Example 1
Add or subtract.
a. b.
Solution a. b.
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81. Copyright © 2011 Pearson Education, Inc.
continued
Simplify.
c. d.
Solution c.
Combine the like radicals by subtracting the coefficients and keeping the radical. d.
Regroup the terms.
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82. Copyright © 2011 Pearson Education, Inc.
Example 2
Add or subtract.
a. b.
Solution a.
Factor 28.
Simplify.
Combine like radicals. b.
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83. Copyright © 2011 Pearson Education, Inc.
continued c.
Solution c.
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84. Copyright © 2011 Pearson Education, Inc.
Example 3a
Find the product.
Solution
Use the distributive property.
Multiply.
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85. Copyright © 2011 Pearson Education, Inc.
Example 3c
Find the product.
Solution
Use the distributive property.
Use the product rule.
Find the products.
Combine like radicals.
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Example 3d
Find the product.
Solution
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87. Copyright © 2011 Pearson Education, Inc.
Example 3e
Find the product.
Solution
Use (a – b)2 = a2 – 2ab – b2.
Simplify.
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88. Copyright © 2011 Pearson Education, Inc.
Example 4a
Find the product.
Solution
Use (a + b)(a – b) = a2 – b2.
Simplify.
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89. Copyright © 2011 Pearson Education, Inc.
Example 4b
Find the product.
Solution
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90. Copyright © 2011 Pearson Education, Inc.
Example 5
Simplify. a. b.
Solution b. a.
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91. Copyright © 2011 Pearson Education, Inc.
Simplify. a) b) c) d)
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92. Copyright © 2011 Pearson Education, Inc.
Simplify. a) b) c) d)
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93. Copyright © 2011 Pearson Education, Inc.
Multiply. a) b) c) d)
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94. Copyright © 2011 Pearson Education, Inc.
Multiply. a) b) c) d)
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95. Rationalizing Numerators and Denominators of Radical Expressions
10.5
Rationalizing Numerators and Denominators of Radical Expressions
1. Rationalize denominators.
2. Rationalize denominators that have a sum or difference with a square root term.
3. Rationalize numerators.
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96. Copyright © 2011 Pearson Education, Inc.
Example 1a
Rationalize the denominator.
Solution
Multiply by
Simplify.
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97. Copyright © 2011 Pearson Education, Inc.
Example 1b
Rationalize the denominator.
Use the quotient rule for square roots to separate the numerator and denominator into two radicals.
Solution
Multiply by
Simplify.
Warning: Never divide out factors common to a radicand and a number not under a radical.
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98. Copyright © 2011 Pearson Education, Inc.
Example 1c
Rationalize the denominator.
Solution
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99. Copyright © 2011 Pearson Education, Inc.
Rationalizing Denominators
To rationalize a denominator containing a single nth root, multiply the fraction by a well chosen 1 so that the product’s denominator has a radicand that is a perfect nth power.
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100. Copyright © 2011 Pearson Education, Inc.
Example 2a
Rationalize the denominator. Assume that variables represent positive values.
Solution
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101. Copyright © 2011 Pearson Education, Inc.
Example 2b
Rationalize the denominator. Assume that variables represent positive values.
Solution
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102. Copyright © 2011 Pearson Education, Inc.
Example 2c
Rationalize the denominator. Assume that variables represent positive values.
Solution
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103. Copyright © 2011 Pearson Education, Inc.
Rationalizing a Denominator Containing a Sum or Difference
To rationalize a denominator containing a sum or difference with at least one square root term, multiply the fraction by a 1 whose numerator and denominator are the conjugate of the denominator.
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104. Copyright © 2011 Pearson Education, Inc.
Example 3a
Rationalize the denominator and simplify. Assume variables represent positive values.
Solution
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105. Copyright © 2011 Pearson Education, Inc.
Example 3b
Rationalize the denominator and simplify. Assume variables represent positive values.
Solution
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106. Copyright © 2011 Pearson Education, Inc.
Example 3c
Rationalize the denominator and simplify. Assume variables represent positive values.
Solution
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107. Copyright © 2011 Pearson Education, Inc.
Example 4a
Rationalize the numerator. Assume variables represent positive values.
Solution
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108. Copyright © 2011 Pearson Education, Inc.
Example 4b
Rationalize the numerator. Assume variables represent positive values.
Solution
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109. Copyright © 2011 Pearson Education, Inc.
Rationalize the denominator. a) b) c) d)
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110. Copyright © 2011 Pearson Education, Inc.
Rationalize the denominator. a) b) c) d)
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111. Copyright © 2011 Pearson Education, Inc.
Rationalize the denominator. a) b) c) d)
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112. Copyright © 2011 Pearson Education, Inc.
Rationalize the denominator. a) b) c) d)
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113. Radical Equations and Problem Solving
10.6
Radical Equations and Problem Solving
1. Use the power rule to solve radical equations.
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114. Copyright © 2011 Pearson Education, Inc.
Radical equation: An equation containing at least one radical expression whose radicand has a variable.
Power Rule for Solving Equations
If both sides of an equation are raised to the same integer power, the resulting equation contains all solutions of the original equation and perhaps some solutions that do not solve the original equation. That is, the solutions of the equation a = b are contained among the solutions of an = bn, where n is an integer.
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115. Copyright © 2011 Pearson Education, Inc.
Example 1
Solve.
a. b.
Solution a. b.
Check
Check
True
True
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116. Copyright © 2011 Pearson Education, Inc.
Example 2a Solve.
Solution
Check:
The number 41 checks. The solution is 41.
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117. Copyright © 2011 Pearson Education, Inc.
Example 2b Solve.
Solution
Check:
True. The solution is 4.
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118. Copyright © 2011 Pearson Education, Inc.
Example 2c Solve.
Solution
Check:
False, so 6 is extraneous. This equation has no real number solution.
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119. Copyright © 2011 Pearson Education, Inc.
Example 3a Solve.
Solution
Check:
The number 4 checks. The solution is 4.
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120. Copyright © 2011 Pearson Education, Inc.
Example 4
Solve.
Solution
Square both sides.
Use FOIL.
Subtract x from both sides.
Subtract 7 from both sides.
Factor.
Use the zero-factor theorem.
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121. Copyright © 2011 Pearson Education, Inc.
continued
Checks
True.
False.
Because 2 does not check, it is an extraneous solution. The only solution is 9.
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122. Copyright © 2011 Pearson Education, Inc.
Example 5a
Solve.
Solution
Check
This solution does not check, so it is an extraneous solution. The equation has no real number solution.
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123. Copyright © 2011 Pearson Education, Inc.
Example 5b
Solve
Solution
Check
The solution set is 13.
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124. Copyright © 2011 Pearson Education, Inc.
Example 6
Solve
Solution
Check
There is no solution.
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125. Copyright © 2011 Pearson Education, Inc.
Solving Radical Equations
To solve a radical equation,
1. Isolate the radical if necessary. (If there is more than one radical term, isolate one of the radical terms.)
2. Raise both sides of the equation to the same power as the root index of the isolated radical.
3. If all radicals have been eliminated, solve. If a radical term remains, isolate that radical term and raise both sides to the same power as its root index.
4. Check each solution. Any apparent solution that does not check is an extraneous solution.
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126. Copyright © 2011 Pearson Education, Inc.
Solve.
a) 6
b) 8
c) 9
d) no solution
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127. Copyright © 2011 Pearson Education, Inc.
Solve.
a) 6
b) 8
c) 9
d) no solution
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128. Copyright © 2011 Pearson Education, Inc.
Solve.
a) 2
b) 4 c)
d) no real-number solution
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129. Copyright © 2011 Pearson Education, Inc.
Solve.
a) 2
b) 4 c)
d) no real-number solution
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130. Copyright © 2011 Pearson Education, Inc.
Solve.
a) 3, 4
b) 3
c) 4
d) no real-number solution
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131. Copyright © 2011 Pearson Education, Inc.
Solve.
a) 3, 4
b) 3
c) 4
d) no real-number solution
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132. Copyright © 2011 Pearson Education, Inc.
10.7
Complex Numbers
1. Write imaginary numbers using i.
2. Perform arithmetic operations with complex numbers.
3. Raise i to powers.
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133. Copyright © 2011 Pearson Education, Inc.
Imaginary unit: The number represented by i, where
and i2 = 1.
Imaginary number: A number that can be expressed in the form bi, where b is a real number and i is the imaginary unit.
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134. Copyright © 2011 Pearson Education, Inc.
Example 1
Write each imaginary number as a product of a real number and i.
a. b. c.
Solution
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135. Copyright © 2011 Pearson Education, Inc.
Rewriting Imaginary Numbers
To write an imaginary number in terms of the imaginary unit i,
1. Separate the radical into two factors,
2. Replace with i.
3. Simplify
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136. Copyright © 2011 Pearson Education, Inc.
Complex number: A number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
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137. Copyright © 2011 Pearson Education, Inc.
Example 2a
Add or subtract. (9 + 6i) + (6 – 13i)
Solution
We add complex numbers just like we add polynomials—by combining like terms.
(9 + 6i) + (6 – 13i) = (9 + 6) + (6i – 13i )
= –3 – 7i
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138. Copyright © 2011 Pearson Education, Inc.
Example 2b
Add or subtract. (3 + 4i) – (4 – 12i)
Solution
We subtract complex numbers just like we subtract polynomials.
(3 + 4i) – (4 – 12i) = (3 + 4i) + (4 + 12i)
= 7 + 16i
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139. Copyright © 2011 Pearson Education, Inc.
Example 3
Multiply.
a. (8i)(4i) b. (6i)(3 – 2i)
Solution
a. (8i)(4i) b. (6i)(3 – 2i)
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140. Copyright © 2011 Pearson Education, Inc.
continued
Multiply.
c. (9 – 4i)(3 + i) d. (7 – 2i)(7 + 2i)
Solution
c. (9 – 4i)(3 + i) d. (7 – 2i)(7 + 2i)
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141. Copyright © 2011 Pearson Education, Inc.
Complex conjugate: The complex conjugate of a complex number a + bi is a – bi.
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142. Copyright © 2011 Pearson Education, Inc.
Example 4a
Divide. Write in standard form.
Solution Rationalize the denominator.
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143. Copyright © 2011 Pearson Education, Inc.
Example 4b
Divide. Write in standard form.
Solution Rationalize the denominator.
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144. Copyright © 2011 Pearson Education, Inc.
Example 5
Simplify.
Solution
= 1
Write i40 as (i4)10.
Write i32 as (i4)8.
Replace i4 with 1.
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145. Copyright © 2011 Pearson Education, Inc.
Simplify. (4 + 7i) – (2 + i)
a) i2
b) i
c) 6 + 6i
d) 6 + 8i
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146. Copyright © 2011 Pearson Education, Inc.
Simplify. (4 + 7i) – (2 + i)
a) i2
b) i
c) 6 + 6i
d) 6 + 8i
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147. Copyright © 2011 Pearson Education, Inc.
Multiply. (4 + 7i)(2 + i)
a)  i
b) i
c)  i
d) i
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148. Copyright © 2011 Pearson Education, Inc.
Multiply. (4 + 7i)(2 + i)
a)  i
b) i
c)  i
d) i
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149. Copyright © 2011 Pearson Education, Inc.
Write in standard form. a) b) c) d)
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150. Copyright © 2011 Pearson Education, Inc.
Write in standard form. a) b) c) d)
Copyright © 2011 Pearson Education, Inc.