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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational Exponents, Radicals, and Complex Numbers CHAPTER 10.1Radical Expressions and Functions 10.2Rational Exponents 10.3Multiplying, Dividing, and Simplifying Radicals 10.4Adding, Subtracting, and Multiplying Radical Expressions 10.5Rationalizing Numerators and Denominators of Radical Expressions 10.6Radical Equations and Problem Solving 10.7Complex Numbers 10

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Radical Expressions and Functions 10.1 1.Find the n th 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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Slide 10- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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 b n = 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 b n = a. nth root: The number b is an nth root of a number a if b n = a.

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Slide 10- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Evaluate each square root. a. b. c. Solution

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Slide 10- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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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Slide 10- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Approximate the roots using a calculator or table in the endpapers. Round to three decimal places. a. b. Solution

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Slide 10- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the root. Assume variables represent nonnegative values. a. b. c. Solution Because (y 2 ) 2 = y 4. Because (6m 3 ) 2 = 36m 6. Solution Because (2x 4 ) 5 = 32x 20.

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Slide 10- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the root. Assume variables represent any real number. a. b. c. Solution

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Slide 10- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Radical function: A function of the form f(x) = where P is a polynomial. Example Solution Given f(x) = find f(3). To find f(3), substitute 3 for x and simplify.

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Slide 10- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the domain of each of the following. a. b. Solution Since the index is even, the radicand must be nonnegative. 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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Slide 10- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 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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Slide 10- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Answer It takes an object 7.071 seconds to fall 800 feet. Check We can verify the calculations, which we will leave to the viewer.

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Slide 10- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley For which square root is –12.37 the approximation for? a) b) c) d)

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Slide 10- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley For which square root is –12.37 the approximation for? a) b) c) d)

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Slide 10- 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate. a) 0.2 b) 0.02 c) 0.002 d) 0.0002

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Slide 10- 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate. a) 0.2 b) 0.02 c) 0.002 d) 0.0002

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Slide 10- 18 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the domain of f(x) =. a) b) c) d)

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Slide 10- 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Find the domain of f(x) =. a) b) c) d)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational Exponents10.2 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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Slide 10- 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational exponents: An exponent that is a fraction. Rational Exponents with a Numerator of 1 a 1/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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Slide 10- 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rewrite using radicals, then simplify. a. 49 1/2 b. 125 1/3 c. 64 1/6 Solution a. b. c.

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Slide 10- 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley General Rule for Rational Exponents where a 0 and m and n are natural numbers other than 1.

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Slide 10- 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rewrite using radicals, then simplify, if possible. a. 27 2/3 b. 243 3/4 c. 9 5/2 Solution a. b. c.

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Slide 10- 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Negative Rational Exponents where a 0, and m and n are natural numbers with n 1.

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Slide 10- 26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rewrite using radicals, then simplify, if possible. a. 25 1/2 b. 27 2/3 Solution a. b.

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Slide 10- 27 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write each of the following in exponential form. a. Solution b. a. b.

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Slide 10- 28 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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:a 0 = 1, where a 0. 0 0 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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Slide 10- 29 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 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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Slide 10- 30 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 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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Slide 10- 31 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Perform the indicated operations. Write the result using a radical. Solution b.a. b.

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Slide 10- 32 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 10- 33 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 10- 34 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) 5 b) 25 c) 25 d) 5

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Slide 10- 35 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) 5 b) 25 c) 25 d) 5

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Slide 10- 36 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) 4 b) c) 4 d)

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Slide 10- 37 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) 4 b) c) 4 d)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiplying, Dividing, and Simplifying Radicals 10.3 1.Multiply radical expressions. 2.Divide radical expressions. 3.Use the product rule to simplify radical expressions.

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Slide 10- 39 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Product Rule for Radicals If and are both real numbers, then

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Slide 10- 40 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the product and write the answer in simplest form. Assume all variables represent nonnegative values. a.b. Solution a.b.

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Slide 10- 41 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Raising an nth Root to the nth Power For any nonnegative real number a, Quotient Rule for Radicals If and are both real numbers, then

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Slide 10- 42 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. Assume variables represent nonnegative values. a. Solution b. a.

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Slide 10- 43 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplifying nth Roots 1. Write the radicand as a product of the greatest possible perfect nth power and a number or expression that has no perfect nth power factors. 2. Use the product rule where a is a perfect nth power. 3. Find the nth root of the perfect nth power radicand.

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Slide 10- 44 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. a.b. Solution

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Slide 10- 45 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. Solution The greatest perfect square factor of 32x 5 is 16x 4. Use the product rule of square roots to separate the factors into two radicals. Find the square root of 16x 4 and leave 2x in the radical.

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Slide 10- 46 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify Solution The greatest perfect square factor of 96a 4 b is 16a 4. Use the product rule of square roots to separate the factors into two radicals. Find the square root of 16a 4 and leave 6b in the radical.

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Slide 10- 47 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. Assume all variables represent nonnegative numbers. a) b) c) d)

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Slide 10- 48 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. Assume all variables represent nonnegative numbers. a) b) c) d)

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Slide 10- 49 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 10- 50 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Adding, Subtracting, and Multiplying Radical Expressions 10.4 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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Slide 10- 52 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Like radicals: Radical expressions with identical radicands and identical indexes. Adding Like Radicals To add or subtract like radicals, add or subtract the coefficients and keep the radicals the same.

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Slide 10- 53 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. a. b. Solution a. b. Combine the like radicals by subtracting the coefficients and keeping the radical. Regroup terms.

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Slide 10- 54 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. a. b. Solution a. b. Factor 28. Combine like radicals. Simplify.

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Slide 10- 55 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the product. Solution Use the distributive property. Multiply.

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Slide 10- 56 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the product. Solution Use the product rule. Use the distributive property. Find the products. Combine like radicals.

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Slide 10- 57 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the product. Solution Simplify. Use (a – b) 2 = a 2 – 2ab – b 2.

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Slide 10- 58 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Find the product. Solution Simplify. Use (a + b)(a – b) = a 2 – b 2.

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Slide 10- 59 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. a. Solution a. b.

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Slide 10- 60 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 10- 61 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. a) b) c) d)

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Slide 10- 62 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply. a) b) c) d)

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Slide 10- 63 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply. a) b) c) d)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rationalizing Numerators and Denominators of Radical Expressions 10.5 1.Rationalize denominators. 2.Rationalize denominators that have a sum or difference with a square root term. 3.Rationalize numerators.

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Slide 10- 65 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rationalize the denominator. Solution Simplify. Multiply by

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Slide 10- 66 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rationalize the denominator. Solution Use the quotient rule for square roots to separate the numerator and denominator into two radicals. Multiply by Simplify. Warning: Never divide out factors common to a radicand and a number not under a radical.

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Slide 10- 67 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rationalizing Denominators To rationalize a denominator containing a single nth root, multiply the fraction by a 1 so that the product’s denominator has a radicand that is a perfect nth power.

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Slide 10- 68 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rationalize the denominator. Assume variables represent nonnegative values. Solution

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Slide 10- 69 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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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Slide 10- 70 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rationalize the denominator and simplify. Assume variables represent nonnegative values. Solution

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Slide 10- 71 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Rationalize the numerator. Assume variables represent nonnegative values. Solution

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Slide 10- 72 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rationalize the denominator. a) b) c) d)

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Slide 10- 73 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rationalize the denominator. a) b) c) d)

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Slide 10- 74 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rationalize the denominator. a) b) c) d)

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Slide 10- 75 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rationalize the denominator. a) b) c) d)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Radical Equations and Problem Solving 10.6 1.Use the power rule to solve radical equations.

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Slide 10- 77 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Power Rule If a = b, then a n = b n. Radical equation: An equation containing at least one radical expression whose radicand has a variable.

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Slide 10- 78 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve. a.b. Solution a. Check True b. Check True

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Slide 10- 79 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Example Solve. Check: The number 4 checks. The solution is 4.

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Slide 10- 80 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve. Solution Square both sides. Use FOIL. Subtract x from both sides. Factor. Use the zero-factor theorem. Subtract 7 from both sides.

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Slide 10- 81 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Checks False. True. Because 2 does not check, it is an extraneous solution. The only solution is 9.

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Slide 10- 82 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve. Solution Check This solution does not check, so it is an extraneous solution. The equation has no solution; the solution set is

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Slide 10- 83 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve Solution Check The solution set is {13}.

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Slide 10- 84 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Radical Equations To solve a radical equation: 1.Isolate the radical. (If there is more than one radical term, then isolate one of the radical terms.) 2. Raise both sides of the equation to the same power as the root index. 3. If all radicals have been eliminated, then solve. If a radical term remains, then 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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Slide 10- 85 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. d) no solution a) 6 b) 8 c) 9

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Slide 10- 86 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. d) no solution a) 6 b) 8 c) 9

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Slide 10- 87 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. d) no real-number solution a) 2 b) 4 c)

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Slide 10- 88 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. d) no real-number solution a) 2 b) 4 c)

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Slide 10- 89 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. d) no real-number solution a) {-3, 4} b) {-3} c) {4}

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Slide 10- 90 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve. d) no real-number solution a) {-3, 4} b) {-3} c) {4}

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Complex Numbers 10.7 1.Write imaginary numbers using i. 2.Perform arithmetic operations with complex numbers. 3.Raise i to powers.

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Slide 10- 92 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Imaginary unit: The number represented by i, where and i 2 = 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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Slide 10- 93 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write each imaginary number as a product of a real number and i. a. b.c. Solution a. b.c.

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Slide 10- 94 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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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Slide 10- 95 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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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Slide 10- 96 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example 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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Slide 10- 97 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Multiply. a. (8i)( 4i)b. (9 – 4i)(3 + i) Solution a. (8i)( 4i) b. (9 – 4i)(3 + i)

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Slide 10- 98 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Complex conjugates: The complex conjugate of a complex number a + bi is a – bi.

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Slide 10- 99 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Write in standard form. Solution Rationalize the denominator.

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Slide 10- 100 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify. Solution = 1 Write i 40 as (i 4 ) 10. Write i 32 as (i 4 ) 8. Replace i 4 with 1.

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Slide 10- 101 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. ( 4 + 7i) – (2 + i) a) 2 + 7i 2 b) 2 + 8i c) 6 + 6i d) 6 + 8i

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Slide 10- 102 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. ( 4 + 7i) – (2 + i) a) 2 + 7i 2 b) 2 + 8i c) 6 + 6i d) 6 + 8i

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Slide 10- 103 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply. ( 4 + 7i)(2 + i) a) 15 + 10i b) 1 + 10i c) 15 + 18i d) 15 + 18i

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Slide 10- 104 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply. ( 4 + 7i)(2 + i) a) 15 + 10i b) 1 + 10i c) 15 + 18i d) 15 + 18i

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Slide 10- 105 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write in standard form. a) b) c) d)

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Slide 10- 106 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Write in standard form. a) b) c) d)

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