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Please complete the prerequisite Skills PG 412 #1-12. Chapter 6: Rational Exponents and Radical Functions. Big ideas: Use Rational Exponents Performing function operations and finding inverse functions Solving radical equations. Lesson 1: Evaluate nth Roots and Use Rational Exponents.

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Please complete the prerequisite skills pg 412 1 12

Please complete the prerequisite SkillsPG 412 #1-12


Chapter 6 rational exponents and radical functions

Chapter 6:Rational Exponents and Radical Functions

Big ideas:

Use Rational Exponents

Performing function operations and finding inverse functions

Solving radical equations


Lesson 1 evaluate nth roots and use rational exponents

Lesson 1: Evaluate nth Roots and Use Rational Exponents


Essential question

Essential question

What is the relationship between nth roots and rational exponents?


Vocabulary

VOCABULARY

  • Nth root of a: For an integer n greater than 1, if bn = a, then b is an nth root of a. written as

  • Index of a radical: The integer n, greater than 1, in the expression


Please complete the prerequisite skills pg 412 1 12

a. Because n = 3 is odd and a = –216 < 0, –216 has one real cube root. Because (–6)3= –216, you can write = 3√–216 = –6 or (–216)1/3 = –6.

b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth roots. Because 34 = 81 and (–3)4 = 81, you can write ±4√ 81 =±3

EXAMPLE 1

Find nth roots

Find the indicated real nth root(s) of a.

a. n = 3, a = –216

b. n = 4, a = 81

SOLUTION


Please complete the prerequisite skills pg 412 1 12

1

1

23

323/5

64

( )3

=

(161/2)3

=

43

=

43

=

64

=

=

16

1

1

1

1

1

1

=

=

=

=

( )3

(321/5)3

323/5

32

5

8

23

8

=

=

=

=

EXAMPLE 2

Evaluate expressions with rational exponents

Evaluate (a) 163/2 and (b)32–3/5.

SOLUTION

Radical Form

Rational Exponent Form

a. 163/2

163/2

b. 32–3/5

32–3/5


Please complete the prerequisite skills pg 412 1 12

Keystrokes

Expression

Display

9 1 5

7 3 4

12 3 8

7

c. ( 4 )3 = 73/4

EXAMPLE 3

Approximate roots with a calculator

a. 91/5

1.551845574

b. 123/8

2.539176951

4.303517071


Please complete the prerequisite skills pg 412 1 12

for Examples 1, 2 and 3

GUIDED PRACTICE

Find the indicated real nth root(s) of a.

1. n = 4, a = 625

3. n = 3, a = –64.

SOLUTION

±5

SOLUTION

–4

2.n = 6, a = 64

4. n = 5, a = 243

SOLUTION

±2

SOLUTION

3


Please complete the prerequisite skills pg 412 1 12

1

3

for Examples 1, 2 and 3

GUIDED PRACTICE

Evaluate expressions without using a calculator.

5. 45/2

7. 813/4

27

SOLUTION

32

SOLUTION

6. 9–1/2

8. 17/8

SOLUTION

SOLUTION

1


Please complete the prerequisite skills pg 412 1 12

Expression

10. 64 2/3

11. (4√ 16)5

12. (3√–30)2

for Examples 1, 2 and 3

GUIDED PRACTICE

Evaluate the expression using a calculator. Round the result to two decimal places when appropriate.

9. 42/5

1.74

SOLUTION

SOLUTION

0.06

SOLUTION

32

9.65

SOLUTION


Please complete the prerequisite skills pg 412 1 12

a. 4x5

= 128

x5

=

32

x

=

32

5

x

2

=

EXAMPLE 4

Solve equations using nth roots

Solve the equation.

Divide each side by 4.

Take fifth root of each side.

Simplify.


Please complete the prerequisite skills pg 412 1 12

b. (x – 3)4

= 21

+

x – 3

=

21

4

+

x

=

21

+ 3

4

or

21

+ 3

x

=

x

=

21

+ 3

4

4

5.14

0.86

x

or

x

EXAMPLE 4

Solve equations using nth roots

Take fourth roots of each side.

Add 3 to each side.

Write solutions separately.

Use a calculator.


Essential question1

Essential question

The nth root of a can be written as a to the

What is the relationship between nth roots and rational exponents?


Simplify the expression 4 3 4 8

Simplify the expression:43*48


Lesson 2 apply properties of rational exponents

Lesson 2: Apply Properties of rational exponents


Essential question2

Essential question

How are the properties of rational exponents related to properties of integer exponents?


Vocabulary1

VOCABULARY

  • Simplest form of a radical: A radical with index n is in simplest form if the radicand has no perfect nth powers as factors and any denominator has been rationalized

  • Like radicals: Radical expressions with the same index and radicand


Please complete the prerequisite skills pg 412 1 12

51

5

51/3

51/3

a. 71/4 71/2

b. (61/2 41/3)2

= (61/2)2 (41/3)2

= 6(1/22) 4(1/32)

= 6 42/3

= 61 42/3

1

c. (45 35)–1/5

= [(4 3)5]–1/5

= 12[5 (–1/5)]

=

12

d.

=

2

42

421/3 2

1/3

e.

=

= 7(1/3 2)

6

61/3

EXAMPLE 1

Use properties of exponents

Use the properties of rational exponents to simplify the expression.

= 73/4

= 7(1/4 + 1/2)

= 12 –1

= (125)–1/5

= 5(1 – 1/3)

= 52/3

= 72/3

= (71/3)2


Please complete the prerequisite skills pg 412 1 12

a.

5

80

12

16

18

216

=

12 18

=

=

6

4

4

4

3

3

3

3

80

b.

=

=

=

2

4

5

EXAMPLE 3

Use properties of radicals

Use the properties of radicals to simplify the expression.

Product property

Quotient property


Please complete the prerequisite skills pg 412 1 12

5

5

3

3

3

3

27

5

27

a.

=

=

3

=

3

135

EXAMPLE 4

Write radicals in simplest form

Write the expression in simplest form.

Factor out perfect cube.

Product property

Simplify.


Please complete the prerequisite skills pg 412 1 12

7

7

8

4

8

4

5

5

5

5

5

5

5

5

5

28

28

32

=

b.

=

=

2

EXAMPLE 4

Write radicals in simplest form

Make denominator a perfect fifth power.

Product property

Simplify.


Please complete the prerequisite skills pg 412 1 12

2

2

2

2

2

3

3

3

3

3

(1 + 7)

a.

8

7

+

=

=

4

3

4

4

4

3

3

10

10

54

27

10

10

b.

=

=

+

(81/5)

2

3

(3 – 1)

c.

2

3

=

=

=

2

(81/5)

(81/5)

12

10

2

=

(2 +10)

(81/5)

EXAMPLE 5

Add and subtract like radicals and roots

Simplify the expression.


Please complete the prerequisite skills pg 412 1 12

3

5

3

24

2

3

250

+

40

3

5

4

4

4

27

3

3

3

3

5

5

3

5

2

for Examples 3, 4, and 5

GUIDED PRACTICE

Simplify the expression.

SOLUTION

SOLUTION

SOLUTION

SOLUTION


Please complete the prerequisite skills pg 412 1 12

3

43(y2)3

a.

4y2

=

=

=

3pq4

b.

(27p3q12)1/3

271/3(p3)1/3(q12)1/3

=

=

=

3

4

4

4

3p(3 1/3)q(12 1/3)

n8

43

m4

m4

m

3

3

(y2)3

64y6

14xy 1/3

4

7x1/4y1/3z6

7x(1 – 3/4)y1/3z –(–6)

=

=

c.

=

=

=

n2

2x 3/4 z –6

4

(n2)4

d.

m4

n8

EXAMPLE 6

Simplify expressions involving variables

Simplify the expression. Assume all variables are positive.


Please complete the prerequisite skills pg 412 1 12

5

=

5

4a8b14c5

4a5a3b10b4c5

5

5

a5b10c5

4a3b4

a.

=

=

b.

=

x

5

ab2c

4a3b4

y8

x y

3

x

y

3

=

3

y9

y8

y

EXAMPLE 7

Write variable expressions in simplest form

Write the expression in simplest form. Assume all variables are positive.

Factor out perfect fifth powers.

Product property

Simplify.

Make denominator a perfect cube.

Simplify.


Please complete the prerequisite skills pg 412 1 12

3

xy

=

3

x y

3

=

y9

y3

EXAMPLE 7

Write variable expressions in simplest form

Quotient property

Simplify.


Please complete the prerequisite skills pg 412 1 12

3z)

(12z

1

3

w

w

+

+

5

5

a.

=

=

9z

3

3

2z2

2z5

3

1

4

(3 – 8) xy1/4

–5xy1/4

b.

=

=

3xy1/4

8xy1/4

5

5

5

c.

=

z

=

12

w

w

=

3

3

3

3

3z

2z2

2z2

54z2

2z2

12z

EXAMPLE 8

Add and subtract expressions involving variables

Perform the indicated operation. Assume all variables are positive.


Please complete the prerequisite skills pg 412 1 12

6xy 3/4

3x 1/2 y 1/2

3q3

3

27q9

5

2x1/2y1/4

w

w3

9w5

x10

y5

x2

y

w

2w2

for Examples 6, 7, and 8

GUIDED PRACTICE

Simplify the expression. Assume all variables are positive.

SOLUTION

SOLUTION

SOLUTION

SOLUTION


Essential question3

Essential question

All properties of integer exponents also apply to rational exponents

How are the properties of rational exponents related to properties of integer exponents?


Let f x 3x 5 find f 6

Let f(x) = 3x + 5. Find f(-6)


Lesson 3 perform function operations and composition

Lesson 3Perform Function operations and composition


Essential question4

Essential question

What operations can be performed on a pair of functions to obtain a third function?


Vocabulary2

VOCABULARY

  • Power Function: A function of the form y=axb, where a is a real number and b is a rational number

  • Composition: The composition of a function g with a function f is h(x) = f(f(x)).


Please complete the prerequisite skills pg 412 1 12

a.

f(x) + g(x)

f(x) – g(x)

b.

EXAMPLE 1

Add and subtract functions

Letf (x)= 4x1/2andg(x)=–9x1/2. Find the following.

SOLUTION

f (x) + g(x)

= 4x1/2 + (–9x1/2)

= [4 + (–9)]x1/2

= –5x1/2

SOLUTION

f (x) – g(x)

= [4 – (–9)]x1/2

= 13x1/2

= 4x1/2 – (–9x1/2)


Please complete the prerequisite skills pg 412 1 12

The functions fand geach have the same domain: all nonnegative real numbers. So, the domains of f + gand f – galso consist of all nonnegative real numbers.

c.

the domains of f + gand f – g

EXAMPLE 1

Add and subtract functions

SOLUTION


Please complete the prerequisite skills pg 412 1 12

a.

f (x) g(x)

b.

6x

f (x)

f (x)

g(x)

g(x)

x3/4

f (x) g(x)

=

=

6x1/4

=

6x(1 – 3/4)

EXAMPLE 2

Multiply and divide functions

Let f (x)= 6xand g(x) = x3/4. Find the following.

SOLUTION

= (6x)(x3/4)

= 6x(1 + 3/4)

= 6x7/4

SOLUTION


Please complete the prerequisite skills pg 412 1 12

The domain of f consists of all real numbers, and the domain of gconsists of all nonnegative real numbers. So, the domain of f gconsists of all nonnegative real numbers. Because g(0) = 0, the domain of is restricted to all positive real numbers.

f

g

EXAMPLE 2

Multiply and divide functions

f

g

and

the domains of f

c.

g

SOLUTION


Please complete the prerequisite skills pg 412 1 12

r(m)

s(m)

(6 106)m0.2

=

241m–0.25

=

• Findr(m) s(m).

EXAMPLE 3

Solve a multi-step problem

Rhinos

For a white rhino, heart rate r(in beats per minute) and life span s(in minutes) are related to body mass m(in kilograms) by these functions:

• Explain what this product represents.


Please complete the prerequisite skills pg 412 1 12

Find and simplify r(m) s(m).

r(m) s(m)

=

241m –0.25 [ (6 106)m0.2 ]

241(6 106)m(–0.25 + 0.2)

=

(1446 106)m –0.05

=

(1.446 109)m –0.05

=

EXAMPLE 3

Solve a multi-step problem

SOLUTION

STEP 1

Write product of r(m) and s(m).

Product of powers property

Simplify.

Use scientific notation.


Please complete the prerequisite skills pg 412 1 12

Interpret r(m) s(m).

EXAMPLE 3

Solve a multi-step problem

STEP 2

Multiplying heart rate by life span gives the total number of heartbeats for a white rhino over its entire lifetime.


Please complete the prerequisite skills pg 412 1 12

f (x) + g(x)

f (x) – g(x)

for Examples 1, 2, and 3

GUIDED PRACTICE

Let f (x) = –2x2/3andg(x) = 7x2/3. Find the following.

SOLUTION

f (x) + g(x)

= –2x2/3 + 7x2/3

= 5x2/3

= (–2 + 7)x2/3

SOLUTION

f (x) – g(x)

= –2x2/3 – 7x2/3

= [–2 + ( –7)]x2/3

= –9x2/3


Please complete the prerequisite skills pg 412 1 12

the domains of f + gand f – g

for Examples 1, 2, and 3

GUIDED PRACTICE

SOLUTION

all real numbers; all real numbers


Please complete the prerequisite skills pg 412 1 12

f (x) g(x)

f (x)

g(x)

for Examples 1, 2, and 3

GUIDED PRACTICE

Let f (x) = 3xandg(x) = x1/5. Find the following.

SOLUTION

3x6/5

SOLUTION

3x4/5


Please complete the prerequisite skills pg 412 1 12

the domains off g and

f

g

for Examples 1, 2, and 3

GUIDED PRACTICE

SOLUTION

all real numbers; all real numbers except x=0.


Please complete the prerequisite skills pg 412 1 12

Use the result of Example 3 to find a white rhino’s number of heartbeats over its lifetime if its body mass is 1.7 105kilograms.

about 7.92 108 heartbeats

for Examples 1, 2, and 3

GUIDED PRACTICE

Rhinos

SOLUTION


Essential question5

Essential question

Two functions can be combined by the operations: +, -, x, ÷ and composition

What operations can be performed on a pair of functions to obtain a third function?


Solve x 4y 3 for y

Solve x=4y3 for y


Lesson 4 use inverse functions

Lesson 4: Use inverse Functions


Essential question6

Essential question

How do you find an inverse relation of a given function?


Vocabulary3

VOCABULARY

  • Inverse relation: A relation that interchanges the input and output values of the original relation. The graph of an inverse relation is a reflection of the graph of the original relation, with y=x as the line of reflection

  • Inverse function: An inverse relation that is a function. Functions f and g are inverses provided that f(g(x)) = x and g(f(x)) = x


Please complete the prerequisite skills pg 412 1 12

5

3

x+

=y

1

3

EXAMPLE 1

Find an inverse relation

Find an equation for the inverse of the relation y = 3x – 5.

y = 3x – 5

Write original relation.

x = 3y – 5

Switch x and y.

x + 5 = 3y

Add 5 to each side.

Solve for y. This is the inverse relation.


Please complete the prerequisite skills pg 412 1 12

x

+

Verify thatf(x) = 3x – 5 and f –1(x) =

5

5

1

1

5

5

are inverse functions.

3

3

3

3

3

3

= x

1

1

5

5

x +

x +

f (f –1(x)) =f

f –1(f(x)) =

f –1((3x – 5)

3

3

3

3

(3x – 5) +

=

– 5

= 3

= x –

+

= x

EXAMPLE 2

Verify that functions are inverses

SOLUTION

STEP 1

STEP 2

Show: that f(f –1(x)) = x.

Show: that f –1(f(x)) = x.

= x + 5 – 5


Please complete the prerequisite skills pg 412 1 12

x 1

x + 1

=y

=y

3

2

for Examples 1, 2, and 3

GUIDED PRACTICE

Find the inverse of the given function. Then verify that your result and the original function are inverses.

1. f(x) = x + 4

3. f(x) = –3x – 1

SOLUTION

x – 4 = y

SOLUTION

2. f(x) = 2x – 1

SOLUTION


Please complete the prerequisite skills pg 412 1 12

for Examples 1, 2, and 3

GUIDED PRACTICE

4. Fitness: Use the inverse function in Example 3 to find the length at which the band provides 13pounds of resistance.

48 inches

SOLUTION


Essential question7

Essential question

Write the original equation.

Switch x and y.

Solve for y.

How do you find an inverse relation of a given function?


Expand and solve x 5 2

Expand and solve:(x-5)2


Lesson 6 solve radical equations

Lesson 6: Solve Radical equations


Essential question8

Essential question

Why is it necessary to check every apparent solution of a radical equation in the original equation?


Vocabulary4

VOCABULARY

  • Radical equation: An equation with one or more radicals that have variables in their radicands

  • Extraneous solution: An apparent solution that must be rejects because it does not satisfy the original equation.


Please complete the prerequisite skills pg 412 1 12

2x+7

2x+7

2x+7

Solve 3 = 3.

= 3

3

( )3

3

= 33

2x+7

= 27

2x

= 20

x

= 10

EXAMPLE 1

Solve a radical equation

Write original equation.

Cube each side to eliminate the radical.

Simplify.

Subtract 7 from each side.

Divide each side by 2.


Please complete the prerequisite skills pg 412 1 12

3

27

2(10)+7

3

3

3

= 3

3

?

?

=

=

EXAMPLE 1

Solve a radical equation

CHECK

Check x = 10 in the original equation.

Substitute 10 for x.

Simplify.

Solution checks.


Please complete the prerequisite skills pg 412 1 12

2. ( x+25 ) = 4

3. (23 x –3 ) = 4

for Example 1

GUIDED PRACTICE

Solve equation. Check your solution.

1.3√ x – 9 = –1

x = 512

ANSWER

ANSWER

x = 11

x = –9

ANSWER


Essential question9

Essential question

Raising both sides of an equation to the same power sometimes results in an extraneous solution

Why is it necessary to check every apparent solution of a radical equation in the original equation?


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