# Rational Exponents, Radicals, and Complex Numbers

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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 Copyright © 2011 Pearson Education, Inc.

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. 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. 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 Copyright © 2011 Pearson Education, Inc.

continued Evaluate each root, if possible. d. e. f. Solution Solution Solution Copyright © 2011 Pearson Education, Inc.

continued Evaluate each root, if possible. g. h. Solution Solution 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.” 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 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 Copyright © 2011 Pearson Education, Inc.

continued Find the root. Assume variables represent nonnegative values. e. f. Solution Solution Copyright © 2011 Pearson Education, Inc.

Example 4 Find the root. Assume variables represent any real number. a. b. c. Solution Solution Solution Copyright © 2011 Pearson Education, Inc.

continued Find the root. Assume variables represent any real number. d. e. c. Solution Solution Solution Copyright © 2011 Pearson Education, Inc.

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. 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. 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. Copyright © 2011 Pearson Education, Inc.

continued Answer It takes an object seconds to fall feet. Check We can verify the calculations, which we will leave to the viewer. Copyright © 2011 Pearson Education, Inc.

For which square root is –12.37 the approximation for? a) b) c) d) Copyright © 2011 Pearson Education, Inc.

For which square root is –12.37 the approximation for? a) b) c) d) Copyright © 2011 Pearson Education, Inc.

Evaluate. a) 0.2 b) 0.02 c) d) Copyright © 2011 Pearson Education, Inc.

Evaluate. a) 0.2 b) 0.02 c) d) Copyright © 2011 Pearson Education, Inc.

Find the domain of f(x) = a) b) c) d) Copyright © 2011 Pearson Education, Inc.

Find the domain of f(x) = a) b) c) d) 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. Copyright © 2011 Pearson Education, Inc.

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. 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. Copyright © 2011 Pearson Education, Inc.

continued Rewrite using radicals, then simplify. d. (16)1/ e. 491/2 f. y1/6 Solution d. e. f. Copyright © 2011 Pearson Education, Inc.

continued Rewrite using radicals, then simplify. g. (100x8)1/ h. 9y1/5 i. Solution d. e. f. Copyright © 2011 Pearson Education, Inc.

General Rule for Rational Exponents where a  0 and m and n are natural numbers other than 1. 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. Copyright © 2011 Pearson Education, Inc.

continued Rewrite using radicals, then simplify, if possible. d. e. f. Solution d. e. f. Copyright © 2011 Pearson Education, Inc.

Negative Rational Exponents where a  0, and m and n are natural numbers with n  1. 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. Copyright © 2011 Pearson Education, Inc.

continued Rewrite using radicals; then simplify if possible. c. d. Solution c. Copyright © 2011 Pearson Education, Inc.

Example 4 Write each of the following in exponential form. a. b. Solution a. b. Copyright © 2011 Pearson Education, Inc.

continued Write each of the following in exponential form. c. d. Solution c. d. 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: 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. 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. 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. Copyright © 2011 Pearson Education, Inc.

Example 5d Use the rules of exponents to simplify. Write the answer with positive exponents. Solution Add the exponents. Copyright © 2011 Pearson Education, Inc.

Example 5e Use the rules of exponents to simplify. Write the answer with positive exponents. Solution Copyright © 2011 Pearson Education, Inc.

Example 5f Use the rules of exponents to simplify. Write the answer with positive exponents. Solution Copyright © 2011 Pearson Education, Inc.

Example 5g Use the rules of exponents to simplify. Write the answer with positive exponents. Solution 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 Copyright © 2011 Pearson Education, Inc.

continued Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values. c. Solution Copyright © 2011 Pearson Education, Inc.

Example 7 Perform the indicated operations. Write the result using a radical. a. b. Solution a. b. Copyright © 2011 Pearson Education, Inc.

continued Perform the indicated operations. Write the result using a radical. c. Solution c. Copyright © 2011 Pearson Education, Inc.

Example 8 Write the expression below as a single radical. Assume that all variables represent nonnegative values. Solution Copyright © 2011 Pearson Education, Inc.

Simplify. a) 5 b) 25 c) 25 d) 5 Copyright © 2011 Pearson Education, Inc.

Simplify. a) 5 b) 25 c) 25 d) 5 Copyright © 2011 Pearson Education, Inc.

Simplify. a) 4 b) c) 4 d) Copyright © 2011 Pearson Education, Inc.

Simplify. a) 4 b) c) 4 d) Copyright © 2011 Pearson Education, Inc.

Product Rule for Radicals If both and are real numbers, then Copyright © 2011 Pearson Education, Inc.

Example 1 Find the product and simplify. Assume all variables represent positive values. a. b. Solution Copyright © 2011 Pearson Education, Inc.

continued Find the product and simplify. Assume all variables represent positive values. c. d. Solution Copyright © 2011 Pearson Education, Inc.

continued Find the product and simplify. Assume all variables represent positive values. e. f. Solution Copyright © 2011 Pearson Education, Inc.

continued Find the product and simplify. Assume all variables represent positive values. g. Solution 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 Copyright © 2011 Pearson Education, Inc.

Example 2 Simplify. Assume variables represent positive values. a. Solution b. c. a. c. b. Copyright © 2011 Pearson Education, Inc.

continued Simplify. Assume variables represent positive values. d. Solution e. d. e. 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. Copyright © 2011 Pearson Education, Inc.

Example 3 Simplify. a. b. Solution Solution Copyright © 2011 Pearson Education, Inc.

continued Simplify. c. d. Solution Solution 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. Copyright © 2011 Pearson Education, Inc.

continued Simplify the radical using prime factorization. b. c. b. Solution c. 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. 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. Copyright © 2011 Pearson Education, Inc.

continued Simplify. c. d. Solution Solution 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 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 Copyright © 2011 Pearson Education, Inc.

Simplify. Assume all variables represent nonnegative numbers. a) b) c) d) Copyright © 2011 Pearson Education, Inc.

Simplify. Assume all variables represent nonnegative numbers. a) b) c) d) 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. Copyright © 2011 Pearson Education, Inc.

Example 3a Find the product. Solution Use the distributive property. Multiply. 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. Copyright © 2011 Pearson Education, Inc.

Example 3d Find the product. Solution Copyright © 2011 Pearson Education, Inc.

Example 3e Find the product. Solution Use (a – b)2 = a2 – 2ab – b2. Simplify. Copyright © 2011 Pearson Education, Inc.

Example 4a Find the product. Solution Use (a + b)(a – b) = a2 – b2. Simplify. Copyright © 2011 Pearson Education, Inc.

Example 4b Find the product. Solution Copyright © 2011 Pearson Education, Inc.

Example 5 Simplify. a. b. Solution b. a. Copyright © 2011 Pearson Education, Inc.

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. Copyright © 2011 Pearson Education, Inc.

Example 1a Rationalize the denominator. Solution Multiply by Simplify. 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. Copyright © 2011 Pearson Education, Inc.

Example 1c Rationalize the denominator. Solution 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. Copyright © 2011 Pearson Education, Inc.

Example 2a Rationalize the denominator. Assume that variables represent positive values. Solution Copyright © 2011 Pearson Education, Inc.

Example 2b Rationalize the denominator. Assume that variables represent positive values. Solution Copyright © 2011 Pearson Education, Inc.

Example 2c Rationalize the denominator. Assume that variables represent positive values. Solution 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. Copyright © 2011 Pearson Education, Inc.

Example 3a Rationalize the denominator and simplify. Assume variables represent positive values. Solution Copyright © 2011 Pearson Education, Inc.

Example 3b Rationalize the denominator and simplify. Assume variables represent positive values. Solution Copyright © 2011 Pearson Education, Inc.

Example 3c Rationalize the denominator and simplify. Assume variables represent positive values. Solution Copyright © 2011 Pearson Education, Inc.

Example 4a Rationalize the numerator. Assume variables represent positive values. Solution Copyright © 2011 Pearson Education, Inc.

Example 4b Rationalize the numerator. Assume variables represent positive values. Solution Copyright © 2011 Pearson Education, Inc.

Rationalize the denominator. a) b) c) d) Copyright © 2011 Pearson Education, Inc.

Rationalize the denominator. a) b) c) d) Copyright © 2011 Pearson Education, Inc.

Rationalize the denominator. a) b) c) d) Copyright © 2011 Pearson Education, Inc.

Rationalize the denominator. a) b) c) d) 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. Copyright © 2011 Pearson Education, Inc.

Example 1 Solve. a. b. Solution a. b. Check Check True True Copyright © 2011 Pearson Education, Inc.

Example 2a Solve. Solution Check: The number 41 checks. The solution is 41. Copyright © 2011 Pearson Education, Inc.

Example 2b Solve. Solution Check: True. The solution is 4. Copyright © 2011 Pearson Education, Inc.

Example 2c Solve. Solution Check: False, so 6 is extraneous. This equation has no real number solution. Copyright © 2011 Pearson Education, Inc.

Example 3a Solve. Solution Check: The number 4 checks. The solution is 4. 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. Copyright © 2011 Pearson Education, Inc.

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

Example 5b Solve Solution Check The solution set is 13. Copyright © 2011 Pearson Education, Inc.

Example 6 Solve Solution Check There is no solution. Copyright © 2011 Pearson Education, Inc.

Solve. a) 6 b) 8 c) 9 d) no solution Copyright © 2011 Pearson Education, Inc.

Solve. a) 6 b) 8 c) 9 d) no solution Copyright © 2011 Pearson Education, Inc.

Solve. a) 2 b) 4 c) d) no real-number solution Copyright © 2011 Pearson Education, Inc.

Solve. a) 2 b) 4 c) d) no real-number solution Copyright © 2011 Pearson Education, Inc.

Solve. a) 3, 4 b) 3 c) 4 d) no real-number solution Copyright © 2011 Pearson Education, Inc.

Solve. a) 3, 4 b) 3 c) 4 d) no real-number solution 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. 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. Copyright © 2011 Pearson Education, Inc.

Example 1 Write each imaginary number as a product of a real number and i. a. b. c. Solution 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 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. 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 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 Copyright © 2011 Pearson Education, Inc.

Example 3 Multiply. a. (8i)(4i) b. (6i)(3 – 2i) Solution a. (8i)(4i) b. (6i)(3 – 2i) 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) Copyright © 2011 Pearson Education, Inc.

Complex conjugate: The complex conjugate of a complex number a + bi is a – bi. Copyright © 2011 Pearson Education, Inc.

Example 4a Divide. Write in standard form. Solution Rationalize the denominator. Copyright © 2011 Pearson Education, Inc.

Example 4b Divide. Write in standard form. Solution Rationalize the denominator. Copyright © 2011 Pearson Education, Inc.

Example 5 Simplify. Solution = 1 Write i40 as (i4)10. Write i32 as (i4)8. Replace i4 with 1. Copyright © 2011 Pearson Education, Inc.

Simplify. (4 + 7i) – (2 + i) a) i2 b) i c) 6 + 6i d) 6 + 8i Copyright © 2011 Pearson Education, Inc.

Simplify. (4 + 7i) – (2 + i) a) i2 b) i c) 6 + 6i d) 6 + 8i Copyright © 2011 Pearson Education, Inc.

Multiply. (4 + 7i)(2 + i) a)  i b) i c)  i d) i Copyright © 2011 Pearson Education, Inc.

Multiply. (4 + 7i)(2 + i) a)  i b) i c)  i d) i Copyright © 2011 Pearson Education, Inc.