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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

2. Use the product rule where a is a perfect nth power. 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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

Example 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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example 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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

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

Simplify. a) b) c) d) 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. 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. 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. b. Regroup terms. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

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

Example Find the product. Solution Use (a – b)2 = a2 – 2ab – b2. Simplify. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find the product. Solution Use (a + b)(a – b) = a2 – b2. Simplify. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

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

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

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

Example Rationalize the denominator. Solution Multiply by Simplify. 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. Solution Multiply by Simplify. Warning: Never divide out factors common to a radicand and a number not under a radical. 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. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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. 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 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 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. 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 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Multiply. a. (8i)(4i) b. (9 – 4i)(3 + i) Solution Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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

Example Simplify. Solution = 1 Write i40 as (i4)10. Replace i4 with 1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplify. (4 + 7i) – (2 + i) a) 2 + 7i2 b) 2 + 8i c) 6 + 6i d) 6 + 8i Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplify. (4 + 7i) – (2 + i) a) 2 + 7i2 b) 2 + 8i c) 6 + 6i d) 6 + 8i 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 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 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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

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