Exponent Rules and Dividing Polynomials5.65.6

1. Divide exponential forms with the same base.2. Divide numbers in scientific notation.3. Divide monomials.4. Divide a polynomial by a monomial.

6. Simplify expressions using rules of exponents.

Objective 1

Divide exponential forms with the same base.

Quotient Rule for Exponents

If m and n are integers and a is a real number, where a 0, then .

mm n

n

aa

a

3

7

2

2

2222222222

2

23

7

44

3

72

12

2222222222

2

2

1 1 1

1 1 1

When an equation in one variable is solved the answer is a point on a line.

Divide:Divide:

14

6

j

j3

5

u

u

3

6

m

m

22

35

yx

yx

2u81

j8146 jj

936 mm 91

m yx3

Objective 2

Divide numbers in scientific notation.

Divide and write the result in scientific notation.

7

3

2.34 10

3.6 10

7

3

2.34 10

3.6 10

7 30.65 10

40.65 10

36.5 10

Note: This is not in scientific notation.

When an equation in one variable is solved the answer is a point on a line.

Divide and write the result in scientific notation:Divide and write the result in scientific notation:

2

4

10881

109649

.

.

15

5

1066

10321

.

. 20102 .

211002 .

61035 .

Objective 3

Divide monomials.

Copyright © 2011 Pearson Education, Inc.

Divide.

3 6 2

4 3

18

24

x y z

x y

3 6 2

4 3

18

24 1

x y z

x y

3 4 6 3 218

24 x y z

1 3 23

4 x y z

3 23 1

4 1 1

y z

x3 23

4

y z

x

12418 2

3

6

4

3 z

y

y

x

x

Divide.

3 6 2

4 3

18

24

x y z

x y

3

12418 2

3

6

4

3 z

y

y

x

x

4

1

x

y3

1

xzy

43 23

Dividing Monomials

1. Divide the coefficients or simplify them to fractions in lowest terms. 2. Use the quotient rule for the exponents with like bases. 3. Write the final expression so that all exponents are positive.

When an equation in one variable is solved the answer is a point on a line.

Divide:Divide:

72

45

42

28

qp

rqp

32

45

4

4

y

xy

xyxy 6443

3

3

3

2

q

rp

Objective 4

Divide a polynomial by a monomial.

5274

52

57

54

52

57

54

5274

From arithmetic…

or

Divide.

6 2 3

2

36 9 6

3

x y x y x

x y

6 2 3

2 2 2

36 9 6

3 3 3

x y x y x

x y x y x y

4 212 3x y x

xy

Divide each term in the polynomial by the monomial.

If a, b, and c are real numbers, variables, or expressions with c 0, then

Dividing a Polynomial by a Monomial

Divide each term in the polynomial by the monomial.

.

a b a b

c c c

When an equation in one variable is solved the answer is a point on a line.

Divide:Divide:

2

253

5

1530

xyz

xyzyzx

2

224

4

42816

hk

khhkhk

hk

k 7

4 2

36 32 zx

Objective 6

Simplify expressions using rules of exponents.

Exponents SummaryZero as an exponent: a0 = 1, where a 0.

Negative exponents:

Product rule for exponents:

Quotient rule for exponents:

Raising a power to a power:

Raising a product to a power:

Raising a quotient to a power:

1 1, ,n n

n n

a aa a

m n m na a am

m nn

aa

a

nm mna a

n n nab a b

n na bb a

n

n

na ab b

Use the quotient rule for exponents.

Use the power rule for exponents.205

63

24mm

54

32

2

4

m

m

142

m

20

6

3264mm

2

1

1

m14

Evaluate coefficients.

Reduce coefficients.

4

1

22

z

yx4

4842

z

yx8

4442

x

zy 8

4416

x

zy

86

53

yy

y86

15

yy

y156

8

yy

y

21

8

y

y 13

1

y

When an equation in one variable is solved the answer is a point on a line.

Simplify:Simplify:

pnm

pnm31

2163

37

24

6

3

y

y

3

273

p

nm

1324

1

y

Slide 5- 24Copyright © 2011 Pearson Education, Inc.

Simplify.

a)

b)

c)

d)

25

3

a

a

16

1

a

4

1

a

16a

4a

5.6

Slide 5- 25Copyright © 2011 Pearson Education, Inc.

Simplify.

a)

b)

c)

d)

25

3

a

a

16

1

a

4

1

a

16a

4a

5.6

Slide 5- 26Copyright © 2011 Pearson Education, Inc.

Divide.

a) b)

c) d)

5.6

6

846

3

181518

x

xxx

25

66x

x 22 65

6 xx

265

6 xx

xx

65

6

Slide 5- 27Copyright © 2011 Pearson Education, Inc.

Divide.

a) b)

c) d)

5.6

6

846

3

181518

x

xxx

25

66x

x 22 65

6 xx

265

6 xx

xx

65

6