Section 10.1 Radical Expressions and Functions. f1-152 = = = 236 = 6. 2x 2-14x + 49 = 21x = ƒ x - 7 ƒ

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1 78 CHAPTER 0 Radicals, Radical Functions, and Rational Exponents Chapter 0 Summary Section 0. Radical Expressions and Functions If b a, then b is a square root of a. The principal square root of a, designated a, is the nonnegative number satisfying b a. The negative square root of a is written - a. A square root of a negative number is not a real number. A radical function in x is a function defined by an expression containing a root of x. The domain of a square root function is the set of real numbers for which the radicand is nonnegative. Let fx - x. f f - # - 0 -, not a real number Domain of f: Set the radicand greater than or equal to zero. - x Ú 0 -x Ú - x Domain of f x ƒ x or - q, ] The cube root of a real number a is written a because a b means that b a. - is not a real number. The nth root of a real number a is written n a. The number n is the index. Every real number has one root when n is odd. x - x + 9 x - 7 ƒ x - 7 ƒ The odd nth root of a, n a, is the number b for which b n a. x + x + Every positive real number has two real roots when n is even. An even root of a negative number is not a real number. If n is even, then n a n ƒ a ƒ. If n is odd, then n a n a. Section 0. Rational Exponents a n n a m a n n a m or n a m a - mn a m n 7 A7B # # # # - A7xyB 7xy A B 8 Properties of integer exponents are true for rational exponents. An expression with rational exponents is simplified when no parentheses appear, no powers are raised to powers, each base occurs once, and no negative or zero exponents appear. Simplify: A 8x y - B. 8 A x B A - y B x 9 y - x 9 y Some radical expressions can be simplifed using rational exponents. Rewrite the expression using rational exponents, simplify, and rewrite in radical notation if rational exponents still appear. 9 x x 9 x x x # x x # x x 8 x0 + 0 x0 0 x + ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

2 Summary 79 Section 0. Multiplying and Simplifying Radical Expressions The product rule for radicals can be used to multiply radicals n a # n b n ab. 7x # 0y 7x # 0y 70xy The product rule for radicals can be used to simplify radicals: n ab n a # n b. A radical expression with index n is simplified when its radicand has no factors that are perfect nth powers. To simplify, write the radicand as the product of two factors, one of which is the greatest perfect nth power. Then use the product rule to take the nth root of each factor. If all variables in a radicand are positive, then n a n a. Some radicals can be simplified after multiplication is performed. Simplify: x 7 y. 7 # # x # x # y 9 # y 7x y 9 xy 7x y 9 # xy x y xy Assuming positive variables, multiply and simplify: x y # xy. x y # xy x y y # x y x Section 0. Adding, Subtracting, and Dividing Radical Expressions Like radicals have the same indices and radicands. Like radicals can be added or subtracted using the distributive property. In some cases, radicals can be combined once they have been simplified # - # # - # The quotient rule for radicals can be used to simplify radicals: n a Ab n a n b. 8 8 x - A x x x (x ) x The quotient rule for radicals can be used to divide radicals with the same indices: n a n b n a Ab. Some radicals can be simplified after the division is performed. Assuming a positive variable, divide and simplify: x x - x - - x 7 # # x # x x # x x x. Section 0. Multiplying with More Than One Term and Rationalizing Denominators Radical expressions with more than one term are multiplied in much the same way that polynomials with more than one term are multiplied. A - B 0- A - BA + B F O I L ISBN # # Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

3 70 CHAPTER 0 Radicals, Radical Functions, and Rational Exponents Section 0. Multiplying with More Than One Term and Rationalizing Denominators (continued) Radical expressions that involve the sum and difference of the same two terms are called conjugates. Use A8 + BA8 - B 8 - AB A + BA - B A - B - # to multiply conjugates. - 0 The process of rewriting a radical expression as an equivalent expression without any radicals in the denominator is called rationalizing the denominator. When the denominator contains a single radical with an nth root, multiply the numerator and the denominator by a radical of index n that produces a perfect nth power in the denominator s radicand. Rationalize the denominator: 7 # x x x 7 7 x. x 8x 7 x x If the denominator contains two terms, rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. - # A + B - A B A + B - A + B Section 0. Radical Equations A radical equation is an equation in which the variable occurs in a radicand. Solving Radical Equations Containing nth Roots. Isolate one radical on one side of the equation.. Raise both sides to the nth power.. Solve the resulting equation.. Check proposed solutions in the original equation. Solutions of an equation to an even power that is radicalfree, but not the original equation, are called extraneous solutions. Solve: x + - x. x + x + A x + B Ax + B x + x + x + 0 x - x - 0 x - x - 0 x + x - x + 0 or x - 0 x - x x - Isolate the radical. Square both sides. (A + B) A + AB + B Subtract x + from both sides. Factor out the GCF. Factor completely. Set variable factors equal to zero. Solve for x. Check both proposed solutions. checks, but - is extraneous. The solution is and the solution set is. Section 0.7 Complex Numbers The imaginary unit i is defined as i -, where i -. The set of numbers in the form a + bi is called the set of complex numbers; a is the real part and b is the imaginary part. If b 0, the complex number is a real number. If b Z 0, the complex number is an imaginary number i # - i ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

4 Review Exercises 7 Section 0.7 Complex Numbers (continued) To add or subtract complex numbers, add or subtract their real parts and add or subtract their imaginary parts. - i - 7-0i - i i i - + i To multiply complex numbers, multiply as if they were polynomials. After completing the multiplication, replace i with -. When performing operations with square roots of negative numbers, begin by expressing all square roots in terms of i. Then multiply. F O I L (-i)(+i)8+0i-i-i 8 + 0i - i i - # # 00- i # 0i 0i The complex numbers a + bi and a - bi are conjugates. Conjugates can be multiplied using the formula A + BA - B A - B. The multiplication of conjugates results in a real number. + i - i - i 9 - i To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator in order to obtain a real number in the denominator. This real number becomes the denominator of a and b in the quotient a + bi. + i - i + i # + i - i + i 0 + i + 8i + i - i 0 + i i i i To simplify powers of i, rewrite the expression in terms of i. Simplify: i 7. Then replace i with - and simplify. i 7 i # i i i - i -i -i CHAPTER 0 REVIEW EXERCISES ISBN In Exercises, find the indicated root, or state that the expression is not a real number A In Exercises 7, find the indicated function values for each function. If necessary, round to two decimal places. If the function value is not a real number and does not exist, so state.. fx x - ; f, f, fa b, f 7. gx x - 8; g, g0, g- In Exercises 8 9, find the domain of each square root function. 8. fx x - 9. gx 00 - x In Exercises 0, simplify each expression. Assume that each variable can represent any real number, so include absolute value bars where necessary. 0. x. x +. x - 8x +. x. x. -x + 7 Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

5 7 CHAPTER 0 Radicals, Radical Functions, and Rational Exponents 0. In Exercises 8, use radical notation to rewrite each expression. Simplify, if possible.. xy In Exercises 9 0, rewrite each expression with rational exponents. 9. 7x 0. A9xyB In Exercises, rewrite each expression with a positive rational exponent. Simplify, if possible... xab In Exercises, use properties of rational exponents to simplify each expression.. x # x.. 8x y. In Exercises 7, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation x 9 y 8. x y 9 9. x # x 0... The function fx 0x models the expenditures, fx, in millions of dollars, for the U.S. National Park Service x years after 98. According to this model, what will expenditures be in 0? 0. In Exercises, use the product rule to multiply... 7x x # 7y # x. x - # x -. If fx 7x - x + 7, express the function, f, in simplified form. Assume that x can be any real number. In Exercises 7 9, simplify by factoring. Assume that all variables in a radicand represent positive real numbers. 7. 0x 8. x 8 y 9. In Exercises 0, multiply and simplify, if possible. Assume that all variables in a radicand represent positive real numbers. 0. x # x... x x y z x y # xy x y z # 8xy z 7 x + # x - Ax - y B x x 0. Assume that all variables represent positive real numbers. In Exercises 7, add or subtract as indicated x + xy In Exercises 8 0, simplify using the quotient rule. x A B 00y 0. y B x 0 In Exercises, divide and, if possible, simplify x 7 00x y. x x - y 0. Assume that all variables represent positive real numbers. In Exercises, multiply as indicated. If possible, simplify any radical expressions that appear in the product.. A + B. A 0 - B 7. A 7 - BA7 + B 8. A x - BAy - B 9. A + 8B 0. A - 0B. A 7 + BA7 - B. A7 - BA7 + B In Exercises 7, rationalize each denominator. Simplify, if possible... A 7. x. 9 A y x A x x y x + x In Exercises 7 79, rationalize each numerator. Simplify, if possible. x A 7 y ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

6 Test 7 0. In Exercises 80 8, solve each radical equation. 80. x + 8. x x - + x 8. x - + x + 8. x + x In 007, state tobacco taxes ranged from $0.07 per pack in South Carolina to $.8 in New Jersey. The graph shows the average state cigarette tax per pack from 00 through 007. Average State Cigarette Tax per Pack Average Tax per Pack $.0 $.00 $0.80 $0.0 $0.0 $ Source: Campaign for Tobacco-Free Kids Year The function fx 0.x + 0. models the average state cigarette tax per pack, fx, x years after 00. a. Find and interpret f. Round to two decimal places. Does this underestimate or overestimate the tax displayed by the graph? By how much? b. According to the model, in which year will the average state cigarette tax be $. per pack? 8. Out of a group of 0,000 births, the number of people, fx, surviving to age x is modeled by the function To what age will 0,000 people in the group survive? 7 - i + i 0.7 In Exercises 87 89, express each number in terms of i and simplify, if possible In Exercises 90 99, perform the indicated operation. Write the result in the form a + bi i + - 0i i - 7-7i 9. ii i i7-8i # - + i i - i i i fx x. In Exercises 00 0, simplify each expression. 00. i 0. i CHAPTER 0 TEST Remember to use your Chapter Test Prep Video CD to see the worked-out solutions to the test questions you want to review. ISBN Let fx 8 - x. a. Find f-. b. Find the domain of f.. Evaluate:. Simplify: In Exercises, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. 8. x. x # x In Exercises 9, simplify each expression. Assume that each variable can represent any real number.. 7x Ax - y B. x - 0x + 8. x y 8 9. A - x 0 In Exercises 0 7, perform the indicated operation and, if possible, simplify. Assume that all variables represent positive real numbers. 0. x # 0y. 8x y # xy x + xy x 8. x. A - B. A - BA + B 7. A7 - B In Exercises 8 0, rationalize each denominator. Simplify, if possible. Assume all variables represent positive real numbers A x x Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.

7 7 CHAPTER 0 Radicals, Radical Functions, and Rational Exponents In Exercises, solve each radical equation.. + x - x. x x - 7. x The function fx.9x + 0. models the average height, fx, in inches, of boys who are x months of age, 0 x 0. Find the age at which the average height of boys is 0. inches.. Express in terms of i and simplify: -7. In Exercises 9, perform the indicated operation. Write the result in the form a + bi.. - i - - 9i 7. - i + i # - + i 9. - i 0. Simplify: i. CUMULATIVE REVIEW EXERCISES (CHAPTERS 0) In Exercises, solve each equation, inequality, or system.. x - y + z - x - y - z x + y - z. x - x. x + x + x +. x + + x - x -. x + - x +. Graph the solution set of the system: In Exercises 7, perform the indicated operations. 7. x + y y - x 7. 8x x -, 0 x - x + y 8. y + x 9. x - x - x - 7x 0. x - x - - x , -. 80x - 0x + x x - x + x -, x -. A + BA - B In Exercises 7, factor completely.. x + 0x - 7. x - 8. The amount of light provided by a light bulb varies inversely as the square of the distance from the bulb.the illumination provided is 0 lumens at a distance of 0 feet. How many lumens are provided at a distance of feet? 9. You invested $000 in two accounts paying 7% and 9% annual interest. At the end of the year, the total interest from these investments was $0. How much was invested at each rate? 0. Although there are students enrolled in the college, this is % fewer students than there were enrolled last year. How many students were enrolled last year? ISBN Introductory & Intermediate Algebra for College Students, Third Edition, by Robert Blitzer. Published by Prentice Hall. Copyright 009 by Pearson Education, Inc.