Radical Functions and Rational Exponents Chapter 7 Radical Functions and Rational Exponents

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.

7-1 Roots and Radical Expressions What you’ll learn … To simplify nth roots 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

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 Root For any real numbers a and b, any positive integer n, if an = b, then a is an nth root of b.

24 = 16 2 is 4th root of 16. (-2)4 = 16 -2 is 4th root of 16. x4 = -16 No 4th root of -16. (√10)4 = 100 4th root of 100 is√10. Type of Number Number of Real nth Roots when n is Even Number of Real nth Roots when n is Odd Positive 2 1 Negative None

Example 1a Finding All Real Roots 36 121 Find all square roots of .0001, -1 and . 1 27 Find all cube roots of 0.008, -1000, and .

Example 1b Finding All Real Roots Find all fourth roots of -.0001, 1 and . 16 81 Find all fifth roots of 0, -1, and 32 .

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

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 written The principal fourth root of 16 is 2 because equals . The other fourth root of 16 is written as which equals -2. 4 √16 4 4 √16 √24 4 - √16

Example 2 Finding Roots Find each real number root. √-27 √81 √49 3 4

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 < 0 For any negative real number a, √an = a when n is even. n

Example 3a Simplifying Radical Expressions Simplify each radical expression. √4x6 √a3b6 √x4y8 3 4

Example 3b Simplifying Radical Expressions Simplify each radical expression. √4x2y4 √-27c6 √x8y12 3 4

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 5.5 oz 6.25 oz. d3 4

7-2 Multiplying and Dividing Radical Expressions What you’ll learn … To multiply radical expressions To divide radical expressions 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

* * * * To multiply radicals consider the following: √16 √9 = 4 3 =12 and √16 9 = √144 = 12 * * * Property Multiplying Radical Expressions If √a and √b are real numbers, then √a √b = √ab. n n n n n *

Example 1a Multiplying Radicals Multiply. Simplify if possible. √3 √12 √3 √-9 √4 √ -4 * 3 3 * 4 4 *

Example 2 Simplifying Radical Expressions Simplify each expressions. Assume that all variables are positive. √50x4 √18x4 3√7x3 2√21x3y2 3 *

Example 3 Multiplying Radical Expressions Multiply and simplify. 3√7x3 2√21x3y2 √54x2y3 √5x3y4 * 3 3 *

√b b = = = = To divide radicals consider the following: √36 6 and 36 (6)2 √36 √25 5 25 (5)2 √25 = = = Property Dividing Radical Expressions If √a and √b are real numbers, then √a a √b b n n n = n n

Example 4 Dividing Radicals Multiply. Simplify if possible. √243 √12x4 √27 √3x √1024x15 √4x

To rationalize a denominator of an expression, rewrite it so there are no radicals in any denominator and no denominators in any radical.

Example 5 Rationalizing the Denominator Rationalize the denominator of each expression. 7 √2x3 √4 5 √10xy √6x

7-3 Binomial Radical Expressions What you’ll learn … To add and subtract radical expressions To multiply and divide binomial radical expressions 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

Like radicals are radical expressions that have the same index and the same radicand. To add or subtract like radicals, use the Distributive Property.

Example 1 Adding and Subtracting Radical Expressions 3 3 5 √ x - 3 √ x 4 √ xy + 5 √ xy 4 √ 2 - 5 √ 3 7 √ 5 - 2 √5 2 √ 7 + 3 √ 7 3 4

Example 2 Simplifying Before Adding or Subtracting 6 √ 18 + 4 √ 8 - 3√ 72 √ 50 + 3 √ 32 - 5 √ 18

Example 4 Multiplying Binomial Radical Expressions (3 + 2√ 5 ) ( 2 + 4 √ 5 ) (√ 2 - √ 5 ) 2

Example 5 Multiplying Conjugates (2 + √ 3 ) ( 2 - √ 3 ) (√ 2 - √ 5 ) (√ 2 + √ 5 )

Example 6 Rationalizing a Binomial Radical Denominator 3 + √5 1 - √5 6 + √15 4 - √15

To simplify expressions with rational exponents What you’ll learn … To simplify expressions with rational exponents 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

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 = 25½ 3 √27 = 27⅓ 4 √16 = 161/4

Example 1 Simplifying Expressions with Rational Exponents 1251/3 5½ 2½ 2½ 2½ 8½ P/R = power/root r √x p ( √x )p r * *

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. Example 253/2 = 25(3*1/2) = (253)½ = √253

Example 2 Converting to and from Radical Form y -2.5 y -3/8 √a3 ( √b )2 √x2 5 3

Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator = 0. Property Example am * an = a m+n 8⅓ * 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

Properties of Rational Exponents Let m and n represent rational numbers. Assume that no denominator = 0. Property Example a-m 1 9 -½ 1 1 am 9 ½ 3 am a m-n π3/2 π 3/2-1/2 = π1 = π an π ½ a m am 5 5⅓ 5⅓ b bm 27 27 ⅓ 3 = = = = = ⅓ = =

Example 4 Simplifying Numbers with Rational Exponents (-32)3/5 4 -3.5

Example 5 Writing Expressions in Simplest Form (16y-8) -3/4 (8x15)-1/3

7-5 Solving Radical Equations What you’ll learn … To solve radical equations 2.07 Use equations with radical expressions to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

A radical equation is an equation that has a variable in a radicand or has a variable with a rational exponent. Radical Equation Not a Radical Equation

Steps for Solving a Radical Equation Get radical by itself. Raise both sides to index power. Solve for x. Check.

Example 1 Solving Radical Equations with Index 2 Solve 2 + √3x-2 = 6 √5x+1 – 6 = 0

Example 2 Solving Radical Equations with Rational Exponents Solve 2 (x – 2)2/3 = 50 3(x+1)3/5 = 24

Real World Connection A 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?

Example 4 Checking for Extraneous Solutions Solve √x – 3 + 5 = x √3x + 2 - √2x + 7 = 0

Example 5 Solving Equations with Two Rational Exponents Solve (2x +1)0.5 – (3x+4)0.25 = 0 Solve (x +1)2/3 – (9x+1)1/3 = 0

7-8 Graphing Radical Functions What you’ll learn … Graph radical functions 2.07 Use equations with radical expressions to model and solve problems; justify results. a) Solve using tables, graphs, and algebraic properties.

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.

Example 1 Translating Square Root Functions Vertically y = √x y = √x + 3

Example 2 Translating Square Root Functions Horizontally y = √x y = √x + 3

Example 3 Graphing Square Root Functions y = -√x

Example 4 Graphing Square Root Functions y = -2√x+1 - 3

Real World Connection The 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. 3 2.5 m

Example 6 Graphing Cube Root Functions 3 y = 2√x+3 - 1

Example 7 Transforming Radical Equations y = √4x-12 Rewrite to make it easy to graph using a translation. Describe the graph. y = √8x-24 +3 3 Rewrite to make it easy to graph using a translation. Describe the graph.

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.