Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 8 Rational Exponents, Radicals, and Complex Numbers

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 Rational Exponents, Radicals, and Complex Numbers 8.1Radical Expressions and Functions 8.2Rational Exponents 8.3Multiplying, Dividing, and Simplifying Radicals 8.4Adding, Subtracting, and Multiplying Radical Expressions 8.5Rationalizing Numerators and Denominators of Radical Expressions 8.6Radical Equations and Problem Solving 8.7Complex Numbers CHAPTER 8

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 Radical Equations and Problem Solving 1.Use the power rule to solve radical equations. 8.6

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 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 a n = b n, where n is an integer. Radical equation: An equation containing at least one radical expression whose radicand has a variable.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Example Solve. a.b. Solution a. Check True b. Check True

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 Example Solve. Solution Check: The number 41 checks. The solution is 41.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Example Solve. Solution Check: True. The solution is  4.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 Example Solve. Solution Check: False, so 6 is extraneous. This equation has no real number solution.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Example Solve. Solution Check: The number 4 checks. The solution is 4.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 Example Solve. Solution Square both sides. Use FOIL. Subtract x from both sides. Factor. Use the zero-factor theorem. Subtract 7 from both sides.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 continued Checks False. True. Because 2 does not check, it is an extraneous solution. The only solution is 9.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 Example Solve. Solution Check This solution does not check, so it is an extraneous solution. The equation has no real number solution.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Example Solve Solution Check The solution set is 13.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 Example Solve Solution Check There is no solution.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 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.