Algebra 2B: Chapter 6 Notes 1

Chapter 6: Radical Functions and Rational Exponents

Concept Byte (Review): Properties of Exponents

Recall from Algebra 1, the Properties (Rules) of Exponents.

Property of Exponents: Product of Powers xm xn = xm + n a) (5x7)(x6) b) 3r4(-12r3) c) 6cd5(5c2d2) Property of Exponents: Power of a Power (xm)n = xmn a) (x5)9 b) [(23)3]2 c) (k4)5 Property of Exponents: Power of a Product

(xy)m = xmym

a) (2x2y)4 b) (8g3h2)3(5g2h2)4 c) (9h2x4)2 Dividing Monomials: Property of Exponents: Quotient Rule

nm

n

m

xx

x

a) 4

9

x

x b)

2

83

xy

yx c)

42

95

cx

cx

Algebra 2B: Chapter 6 Notes 2

Zero Property of Exponents

0 1x

a)

0

y

x b)

0

3

2

7

2

y

x c)

t

st 03

Property of Negative Exponents

1m

mx

x

a) 25 b) 4 3

5 8

x y

x y

c) 703 xyx

Property of Exponents: Power of a Quotient

m

mm

y

x

y

x

a)

5

y

x b)

1

32

425

zn

nb c)

3

2

3

Algebra 2B: Chapter 6 Notes 3

Example: Simplify and rewrite each expression using only positive exponents.

1. 3 4(5 )( 3 )a a 2. 3 5 2( 4 )x y

3. 6 3

5 3

4ab c

a bc 4.

12

3

3a b

b

5.

Algebra 2B: Chapter 6 Notes 4

6.1 Roots and Radical Expressions

23 9 We say “3” is the ________ root of 9 and write this:

34 64 We say “4” is the _______ root of 64 and write this:

42 16 We say “2” is the _______ root of 16 and write this:

When we talk about roots:

odd roots 3 5, ,a a etc.

are positive is a is positive and negative if a is negative.

Example: 3 1000 3 8

even roots 64, , ,a a a etc.

are only possible “REAL NUMBERS” if a is positive. We will also only consider the “positive” root/ also called the “principle root.”

Example: 100 4 625 49

Algebra 2B: Chapter 6 Notes 5 Example 1: Example 2: Find the real cube roots of each number. Find the real fourth roots of each number. a) 0.008 a) 16 b) -1000 b) -0.0001

c)

c) 16

81

Example 3: Example 4: Find the real fifth roots of each number. Find the real square roots of each number. a) 0 a) .01 b) -1 b) -25

c) 32 c) 36

121

Example 5: Find each real number root.

a) 3 125 b) 4 81 c) 2

7 d) 27

“Radical Expressions” Or, what happens when variables get involved. ** Note ** in your book, the authors take a lot of time trying to help you understand that when variables are involved, you need to be especially careful about the ideas of positive/negative. The authors really want you to use absolute value signs to “force” a variable to be positive. For the purposes of our class, we will be writing our answers without these absolute value signs.

Algebra 2B: Chapter 6 Notes 6

Example 6: Simplify each radical expression

a) 816x b) 3 6 9a b c) 8 124 x y

d) 481x c) 3 12 15125a b e) 12 16 84 x y z

Example 7:

You can use the expression D = 1.2 h to approximate the visibility range D,in miles, from a height of h feet

above ground. How far above ground is an observer whose visibility range is 84 miles?

Recall from “Rules of Exponents”

3

2x

4( )ab

2

32xy

By this same logic: Simplify each radical expression.

3 6x

4 44 a b

2 64x y

Algebra 2B: Chapter 6 Notes 7

6.2 Multiplying and Dividing Radical Expressions We’ve talked a little about simplifying numerical expressions as we have solved quadratic equations using the quadratic formula and by square roots. We will use these same ideas to multiply radical expressions.

Recall: If and are positive,

then .

a b

a b a b

Write in simplest radical form: 200 75 48

Example 1: Can you simplify the product of the rational expression?

a)

b)

If the radicand has a perfect root among its factors, you can used the product rule to simplify. This is called simplest radical form and is NOT A CALCULATOR ESTIMATE. Example 2: Write in simplest radical form.

a) 3 250 b) 5 160

c) 4 162 d) 3 56

Algebra 2B: Chapter 6 Notes 8 We can still use our “division” idea to deal with variable exponents, but now let’s consider expressions that don’t divide evenly.

Yesterday: 204 a Today: 144 a Think: 4 14

Example 3: Write in simplest radical form.

a) 3 3 8a b b) 7 9x y z c) 95 xy

Example 4: What is the simplest form of the expression?

a) b) 2 3 54 4x y x y

c) d)

Simplifying a Product: Step 1: Use the product rule to combine like radicals Step 2: Simplify using perfect Nth factors.

Algebra 2B: Chapter 6 Notes 9

e) 9 2 3 7x y f) 4 2 3 8x x

g)

h) 2 37 3 2 6y x y

Quotients: Dividing Radicals

Example 5: Simplify the following quotients

a) 5

3

18

2

x

x

b)53

23

162

3

y

y

c)6

4

50

2

x

x

Algebra 2B: Chapter 6 Notes 10

A frequent simplification issue:

14

28

To solve this simplification problem we are going to

RATIONALIZE THE DENOMINATOR! Rationalize the denominator:

Multiply the fraction by something equivalent to 1. (The same value to the top and bottom…)

Goal: Create a perfect square/ perfect nth factor on the denominator. Example 6: Rationalize the denominator of each expression:

a) 6

x

b) 9

2

x

c) 2

3

18

2

x y

y

d) 23

3 2

7

4

xy

x

e) 8

x

y

f) 42

3

4

a

b c

For a radical expression to be “simplified”

No perfect square factors under radicals

No radicals in denominators

No denominators under radicals

Algebra 2B: Chapter 6 Notes 11

6.3 Binomial Radical Expressions “Like Radicals” are radicals with the same index and the same radicands. You can multiply and divide any radicals with the same index. HOWEVER, You can only add and subtract LIKE RADICALS. Be especially cautious when combining/adding radicals.

Example:

3 3 1.73 1.73 3.46

2 3 2(1.73) 3.46

3 3 2 3

2 3 1.41 1.73 3.14

5 2.24

2 3 5

Example 1: What is the simplified form of each expression?

a) 3 5 2 5x x b) 26 7 4 5x x c) 3 512 7 8 7xy xy

d) 37 5 4 5 e) 3 4x xy x xy f) 5 52 217 3 15 3x x

Sometimes you may have like radicals, but you can’t “see” them until you simplify. Example 2: What is the simplest form of the radical expression?

a) 12 75 3 b) 3 3 3250 54 16

Algebra 2B: Chapter 6 Notes 12 Sometimes you have to use FOIL to simplify a radical expression. Example 3: What is the product of each radical expression?

a) 3 2 5 2 4 5 b) 3 7 5 7

c) 6 12 6 12 d) 3 8 3 8

Notice that in parts (c) and (d) that you are multiplying CONJUGATES: a b and a b

Any time you multiple radical conjugates, the result is a rational number. Example: Here our denominator is: _____________ so we want to multiply by its conjugate ______________.

5

2 3

Algebra 2B: Chapter 6 Notes 13 Example 4: Write the expression with a rationalized denominator.

a) 2 7

3 5 b)

4

3 6

x

6.4 Rational Exponents Check the following in your calculator:

1249

12121

1216

13125

1481

1532

Algebra 2B: Chapter 6 Notes 14

mn m na a

Dealing with Rational Exponents: 1. Rewrite the expression as a radical. 2. Multiply if necessary using radical rules. 3. Simplify if you can.

Example 1:

a) 1

264 b) 1 1

2 211 11 c) 1 1

2 23 12 Example 2: Rewrite in simplest radical form

a) 3

7x b) 0.2x

c) 5

8w

d) 3.5y

Example 3: Rewrite in exponential form.

a) 5a b) 5 3b c) 24 x d) 4

5 y

Algebra 2B: Chapter 6 Notes 15

Example 4: What is each product or quotient in simplest form.

a) 34

8 2

x

x

b) 43 3

c) 3

3 2

x

x

d) 37 7

Algebra 2B: Chapter 6 Notes 16 Example 5: What is each number in simplest radical form?

a) 3.59

b) 3

481

c) 3

532

d) 2.516 Example 6: Simplify each expression.

a) 1

15 3(8 )x

b)

Algebra 2B: Chapter 6 Notes 17

6.5 Solving Radical Equations Any time you have a variable under the radical sign, you may have to use exponents to solve.

Steps Solve: 3 2 3 8x

1. Isolate the radical 2. Raise each side of the equation to the nth powers 3. Solve the equation 4. CHECK YOUR ANSWERS!!!! Example 1:

a) 4 1 5 0x b) 5 1 3x x

c) 353 ( 1) 1 25x d) 2

33( 1) 12x

Algebra 2B: Chapter 6 Notes 18

e) 3 1 2 7 0x x f) 2 2 10 4x

6.8 Graphing Radical Functions Graph the following radical expression

y x 3y x

x

y

x

y

Algebra 2B: Chapter 6 Notes 19

2y x 2 1y x

3 4y x 3 4y x

x

y

x

y

x

y

x

y