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 7-1/7.2 Roots and Radical Expressions What you’ll learn …To simplify nth roots
4 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
5 Example 2 Finding RootsFind each real number root.√16√81√124
6 √-27 √64 √-16 Find each real number root. Example 2 Finding Roots 3 3
7 Example 3a Simplifying Radical Expressions Simplify each radical expression.√4x6√a3b6√x4y834
8 Example 3b Simplifying Radical Expressions Simplify each radical expression.√4x2y4√-27c6√x8y1234
9 * * * * 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*
10 Example 1a Multiplying Radicals Multiply. Simplify if possible.√3 √12√3 √-9√4 √ -4*33*44*
11 Example 2 Simplifying Radical Expressions Simplify each expressions. Assume that all variables are positive.√50x4√18x43√7x3 2√21x3y23*
12 Example 3 Multiplying Radical Expressions Multiply and simplify.3√7x √21x3y2√54x2y3 √5x3y4*33*
13 √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
14 Example 4 Dividing Radicals Multiply. Simplify if possible.√ √12x4√ √3x√1024x15√4x
15 To rationalize a denominator of an expression, rewrite it so there are no radicals in any denominator and no denominators in any radical.
16 Example 5 Rationalizing the Denominator Rationalize the denominator of each expression.√2x √4√10xy √6x
17 7-3 Binomial Radical Expressions What you’ll learn …To add and subtract radical expressionsTo multiply and divide binomial radical expressions
18 Like radicals are radical expressions that have the same index and the same radicand.
19 Example 1 Adding and Subtracting Radical Expressions 335 √ x √ x √ xy + 5 √ xy4 √ √ √ √52 √ √ 734
20 Example 2 Simplifying Before Adding or Subtracting 6 √ √ √ 72√ √ √ 18
23 Example 6 Rationalizing a Binomial Radical Denominator 3 + √51 - √56 + √154 - √15
24 To simplify expressions with rational exponents What you’ll learn …To simplify expressions with rational exponents
25 Another way to write a radical expression is to use a rational (fractional) exponent. Like the radical form, the exponent form always indicates the principal root.√25 = 25½3√27 = 27⅓4√16 = 161/4
26 Example 1 Simplifying Expressions with Rational Exponents 1251/35½2½ ½2½ ½P/R = power/rootr√x p( √x )pr**
27 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
28 Example 2 Converting to and from Radical Form y -2.5y -3/8√a3( √b )2√x253
29 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
30 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==
31 Example 4 Simplifying Numbers with Rational Exponents (-32)3/54 -3.5
32 Example 5 Writing Expressions in Simplest Form (16y-8) -3/4(8x15)-1/3
33 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.
34 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
35 Steps for Solving a Radical Equation Get radical by itself.Raise both sides to index power.Solve for x.Check.
36 Example 1 Solving Radical Equations with Index 2 Solve2 + √3x-2 = 6√5x+1 – 6 = 0
37 Example 2 Solving Radical Equations with Rational Exponents Solve2 (x – 2)2/3 = 503(x+1)3/5 = 24
38 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?
39 Example 4 Checking for Extraneous Solutions Solve√x – = x√3x √2x + 7 = 0
40 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
41 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.
42 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.
43 Example 1 Translating Square Root Functions Vertically y = √xy = √x + 3
44 Example 2 Translating Square Root Functions Horizontally y = √xy = √x + 3
45 Example 3 Graphing Square Root Functions y = -√x
46 Example 4 Graphing Square Root Functions y = -2√x
47 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
48 Example 6 Graphing Cube Root Functions 3y = 2√x
49 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.
50 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.