Presentation on theme: "Radical Functions and Rational Exponents"— Presentation transcript:
1 Radical Functions and Rational Exponents Chapter 7Radical Functions and Rational Exponents
2 In this chapter, you will … You will extend your knowledge of roots to include cube roots, fourth roots, fifth roots, and so on.You will learn to add, subtract, multiply, and divide radical expressions, including binomial radical expressions.You will solve radical equations, and graph translations of radical functions and their inverses.
3 7-1 Roots and Radical Expressions What you’ll learn …To simplify nth roots1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.
4 Since 52 = 25, 5 is a square root of 25. Since 53 = 125, 5 is a cube root of 125.Since 54 = 625, 5 is a fourth root of 625.Since 55 = 3125, 5 is a fifth root of 3125.This pattern leads to the definition of nth root.Definition nth RootFor any real numbers a and b, any positive integer n, if an = b, then a is an nth root of b.
5 24 = 16 2 is 4th root of 16. (-2)4 = 16 -2 is 4th root of 16. x4 = No 4th root of -16.(√10)4 = th root of 100 is√10.Type of NumberNumber of Real nth Roots when n is EvenNumber of Real nth Roots when n is OddPositive21NegativeNone
6 Example 1a Finding All Real Roots 36121Find all square roots of .0001, -1 and127Find all cube roots of 0.008, -1000, and
7 Example 1b Finding All Real Roots Find all fourth roots of , 1 and1681Find all fifth roots of 0, -1, and 32 .
8 A radical sign is used to indicate a root. The number under the radical sign is the radicand.The index gives the degree of the root.radical sign
9 When a number has two real roots, the positive root is called the principal root and the radical sign indicates the principal root. The principal fourth root of 16 is writtenThe principal fourth root of 16 is 2 becauseequals The other fourth root of 16 is written as which equals -2.4√1644√16√244- √16
10 Example 2 Finding RootsFind each real number root.√-27√81√4934
11 For any negative real number a, √an = a when n is even. Notice that when x=5, √x2 = √52 = √25 = 5 =x.And when x=-5, √x2 = √(-5)2 = √25 = 5 ≠ x.Property nth Root of an, a < 0For any negative real number a,√an = a when n is even.n
12 Example 3a Simplifying Radical Expressions Simplify each radical expression.√4x6√a3b6√x4y834
13 Example 3b Simplifying Radical Expressions Simplify each radical expression.√4x2y4√-27c6√x8y1234
14 Example 4 Real World Connection A citrus grower wants to ship a select grade of oranges that weigh from 8 to 9 ounces in gift cartons. Each carton will hold three dozen oranges, in 3 layers of 3 oranges by 4 oranges.The weight of an orange is related to its diameter by the formula w = , where d is the diameter in inches and w is the weight in ounces. Cartons can be ordered in whole inch dimensions. What size cartons should the grower order?Find the diameter if w = 3 oz oz oz.d34
15 7-2 Multiplying and Dividing Radical Expressions What you’ll learn …To multiply radical expressionsTo divide radical expressions1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.
16 * * * * To multiply radicals consider the following: √16 √9 = 4 3 =12 and √16 9 = √144 = 12***Property Multiplying Radical ExpressionsIf √a and √b are real numbers, then √a √b = √ab.nnnnn*
17 Example 1a Multiplying Radicals Multiply. Simplify if possible.√3 √12√3 √-9√4 √ -4*33*44*
18 Example 2 Simplifying Radical Expressions Simplify each expressions. Assume that all variables are positive.√50x4√18x43√7x3 2√21x3y23*
19 Example 3 Multiplying Radical Expressions Multiply and simplify.3√7x √21x3y2√54x2y3 √5x3y4*33*
20 √b b = = = = To divide radicals consider the following: √ and (6) √36√ (5) √25===Property Dividing Radical ExpressionsIf √a and √b are real numbers, then √a a√b bnnn=nn
21 Example 4 Dividing Radicals Multiply. Simplify if possible.√ √12x4√ √3x√1024x15√4x
22 To rationalize a denominator of an expression, rewrite it so there are no radicals in any denominator and no denominators in any radical.
23 Example 5 Rationalizing the Denominator Rationalize the denominator of each expression.√2x √4√10xy √6x
24 7-3 Binomial Radical Expressions What you’ll learn …To add and subtract radical expressionsTo multiply and divide binomial radical expressions1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.
25 Like radicals are radical expressions that have the same index and the same radicand. To add or subtract like radicals, use the Distributive Property.
26 Example 1 Adding and Subtracting Radical Expressions 335 √ x √ x √ xy + 5 √ xy4 √ √ √ √52 √ √ 734
27 Example 2 Simplifying Before Adding or Subtracting 6 √ √ √ 72√ √ √ 18
30 Example 6 Rationalizing a Binomial Radical Denominator 3 + √51 - √56 + √154 - √15
31 To simplify expressions with rational exponents What you’ll learn …To simplify expressions with rational exponents1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.
32 Another way to write a radical expression is to use a rational exponent. Like the radical form, the exponent form always indicates the principal root.√25 = 25½3√27 = 27⅓4√16 = 161/4
33 Example 1 Simplifying Expressions with Rational Exponents 1251/35½2½ ½2½ ½P/R = power/rootr√x p( √x )pr**
34 A rational exponent may have a numerator other than 1 A rational exponent may have a numerator other than 1. The property (am)n = amn shows how to rewrite an expression with an exponent that is an improper fraction.Example253/2 = 25(3*1/2) = (253)½ = √253
35 Example 2 Converting to and from Radical Form y -2.5y -3/8√a3( √b )2√x253
36 Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator = 0.Property Exampleam * an = a m+n ⅓ * 8⅔ = 8 ⅓+⅔ = 81 =8(am)n = amn (5½)4 = 5½*4 = 52 = 25(ab)m = ambm (4 *5)½ = 4½ * 5½ =2 * 5½
37 Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator = 0.Property Examplea-m ½am ½am a m-n π3/ π 3/2-1/2 = π1 = πan π ½a m am ⅓ ⅓b bm ⅓=====⅓==
38 Example 4 Simplifying Numbers with Rational Exponents (-32)3/54 -3.5
39 Example 5 Writing Expressions in Simplest Form (16y-8) -3/4(8x15)-1/3
40 7-5 Solving Radical Equations What you’ll learn …To solve radical equations2.07 Use equations with radical expressions to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.
41 A radical equation is an equation that has a variable in a radicand or has a variable with a rational exponent.Radical EquationNot a Radical Equation
42 Steps for Solving a Radical Equation Get radical by itself.Raise both sides to index power.Solve for x.Check.
43 Example 1 Solving Radical Equations with Index 2 Solve2 + √3x-2 = 6√5x+1 – 6 = 0
44 Example 2 Solving Radical Equations with Rational Exponents Solve2 (x – 2)2/3 = 503(x+1)3/5 = 24
45 Real World ConnectionA company manufactures solar cells that produce 0.02 watts of power per square centimeter of surface area. A circular solar cell needs to produce at least 10 watts. What is the minimum radius?
46 Example 4 Checking for Extraneous Solutions Solve√x – = x√3x √2x + 7 = 0
47 Example 5 Solving Equations with Two Rational Exponents Solve(2x +1)0.5 – (3x+4)0.25 = 0Solve(x +1)2/3 – (9x+1)1/3 = 0
48 7-8 Graphing Radical Functions What you’ll learn …Graph radical functions2.07 Use equations with radical expressions to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.
49 A radical equation defines a radical function A radical equation defines a radical function. The graph of the radical function y= √x + k is a translation of the graph of y= √x. If k is positive, the graph is translated k units up. If k is negative, the graph is translated k units down.
50 Example 1 Translating Square Root Functions Vertically y = √xy = √x + 3
51 Example 2 Translating Square Root Functions Horizontally y = √xy = √x + 3
52 Example 3 Graphing Square Root Functions y = -√x
53 Example 4 Graphing Square Root Functions y = -2√x
54 Real World ConnectionThe function h(x) = 0.4 √ x models the height h in meters of a female giraffe that has a mass of x kilograms. Graph the model with a graphing calculator. Use the graph to estimate the mass of the young giraffe in the photo.32.5 m
55 Example 6 Graphing Cube Root Functions 3y = 2√x
56 Example 7 Transforming Radical Equations y = √4x-12Rewrite to make it easy to graph using a translation. Describe the graph.y = √8x3Rewrite to make it easy to graph using a translation. Describe the graph.
57 In this chapter, you should have … Extended your knowledge of roots to include cube roots, fourth roots, fifth roots, and so on.Learned to add, subtract, multiply, and divide radical expressions, including binomial radical expressions.Solved radical equations, and graphed translations of radical functions and their inverses.