the product, quotient, and power rules for exponents...quotient rule for exponents example: p 8 p3 =...

The Product,

Quotient, and Power

Rules for Exponents

OBJECTIVES

Multiply expressions

using the product rule

for exponents.

A

OBJECTIVES

Divide expressions

using the quotient rule

for exponents.

B

OBJECTIVES

Use the power rules to

simplify expressions.

C

RULES

Signs for Multiplication

1. When multiplying two

numbers with the same

sign, product is positive (+).

RULES

Signs for Multiplication

2. When multiplying two

numbers with different signs,

product is negative (-).

RULES

Signs for Division

1.When dividing two

numbers with the same

sign, product is positive (+).

RULES

Signs for Division

2.When dividing two numbers

with different signs, product

is negative (-).

RULES FOR EXPONENTS

If m, n, and k are positive

integers, then:

1. Product rule for exponents

xmxn = xm+n

Example:

x5•x6 = x5+6 = x11

RULES FOR EXPONENTS


integers, then:

2. Quotient rule for exponents

- > , 0=m m nn m n xx x

x

RULES FOR EXPONENTS


integers, then:

2. Quotient rule for exponents

Example:

p8

p3= p8-3 = p5

RULES FOR EXPONENTS


integers, then:

3. Power rule for products

=k

mk nkm n yy xx

RULES FOR EXPONENTS


integers, then:

3. Power rule for products

Example:

= =4

4 3 4 4 3 4 16 12x y x xy y• •

RULES FOR EXPONENTS


integers, then:

4. Power rule for quotients

0=m m

myx x

y y

RULES FOR EXPONENTS


integers, then:

4. Power rule for quotients

Example: 6

= =3 3 6 18

4 4 6 24a a a

b b b

•

•

Section 4.1

Exercise #1

Chapter 4

Exponents and Polynomials

Find.

a. (2a3b)(– 6ab3 )

= (2 • – 6)a3+1 b1+3

= – 12a4b4

b. (– 2x2yz)(– 6xy3z 4)

= ( – 2 • – 6)x2 + 1 y1 + 3 z1 + 4

= 12x3y5z5

Find.

c. 18x5y7

– 9xy3

=

18

– 9

x5 – 1 y7 – 3

= – 2x4y4

Section 4.1

Exercise #2

Chapter 4


Find.

3 2 3 3 3 3 2 3(2 ) = 2x y x y

= 8x9y6

b. ( – 3x2y3 )2

a. (2x3y2 )3

2 3 2 2 2 2 3 2( – 3 ) = ( – 3) x y x y

= 9x4y6

Section 4.2

Integer Exponents

OBJECTIVES

Write an expression with

negative exponents as

an equivalent one with

positive exponents.

A

OBJECTIVES

Write a fraction

involving exponents as

a number with a

negative power.

B

OBJECTIVES

Multiply and divide

expressions involving

negative exponents.

C

RULES

Zero Exponent 0For 0, =1x x

– n 1= 0nx xx

If n is a positive integer,

Negative Exponent

RULES

nth Power of a Quotient

–1 =

nnx

x

RULES

x–m

y–n = yn

xm

For any nonzero numbers

x and y and any positive

integers m and n:

Simplifying Fractions with

Negative Exponents

Section 4.2

Exercise #4

Chapter 4


Simplify and write the answer without negative

exponents.

– 7

1a. x

– 7– 1 = x

= x( – 1) ( – 7 )

= x 7

Simplify and write the answer without negative

exponents.

b. x – 6

x – 6

= x – 6 – – 6

0 = = 1, 0xx

= x – 6 + 6

Section 4.2

Exercise #5

Chapter 4


Simplify.

– 3 4

2 3

– 2

2

b. 3

x y

x y

= 2 –2 x – 3 –2

y 4 –2

3–2 x

2 –2 y

3 –2

= 2 –2 x 6 y –8

3–2 x

– 4 y

–6

= 32 x 6 – – 4

y –8 –(–6)

22

=

9 x10 y –2

4

= 9 x10

4y2

Simplify.

= 2 – 2 3 – 1( – 2) x – 5( – 2) y ( – 2)

= 2 – 2 3 2 x 10 y – 2

2 102 21 1 = 3

2x

y

= 9x10

4y2

Section 4.3

Application

of Exponents:

Scientific Notation

OBJECTIVES

Write numbers in

scientific notation.

A

OBJECTIVES

Multiply and divide

numbers in scientific

notation.

B

Solve applications. C

RULES

M10n

A number in scientific notation

is written as

Where M is a number between

1 and 10 and n is an integer.

PROCEDURE

1. Move decimal point in

number so there is only

one nonzero digit to its

left.

(M10n)

The resulting number is M.

Writing a number in scientific

notation

PROCEDURE

2. If the decimal point is moved

to the left, n is positive;

(M10n)


notation

If the decimal point is moved

to the right, n is negative.

PROCEDURE

3. Write (M10n).

(M10n)


notation

PROCEDURE

Multiplying using scientific

notation

1. Multiply decimal parts first.

Write result in scientific

notation.

PROCEDURE


notation

2. Multiply powers of 10

using product rule.

PROCEDURE


notation

3. Answer is product

obtained in steps 1 and 2

after simplification.

Section 4.3

Exercise #6

Chapter 4


a. 48,000,000

Write in scientific notation.

= 4 8000000 .

= 4.8107

b. 0.00000037

= 0.0000003 7

= 3.7 10 – 7

Section 4.3

Exercise #7

Chapter 4


Perform the indicated operations.

4 6a. 3 10 7.1 10

4 + 6 = 3 7.1 10

= 21.3 1010

= 2.13 101 + 10

= 2.13 1011

= 2.13 101 1010

Section 4.4

Polynomials:

An Introduction

OBJECTIVES

Classify polynomials. A

Find the degree of a

polynomial.

B

OBJECTIVES

Write a polynomial in

descending order.

C

Evaluate polynomials. D

DEFINITION

Polynomial

An algebraic expression

formed using addition and

subtraction on products of

numbers and variables raised

to whole number exponents.

Section 4.4

Exercise #8

Chapter 4


Classify as a monomial (M), binomial (B), or trinomial (T).

a. 3x – 5

B, binomial

b. 5x3

M, monomial

c. 8x2 – 2 + 5x

T, trinomial

Section 4.4

Exercise #10

Chapter 4


Find the value.

– 16t2 + 100 when t = 2

= – 16(2)2 + 100

= – 16(4) + 100

= – 64 + 100

= 36

Section 4.5

Addition and Subtraction

of Polynomials

OBJECTIVES

Add polynomials. A

Subtract polynomials. B

OBJECTIVES

Find areas by adding

polynomials.

C

Solve applications. D

Section 4.5

Exercise #11

Chapter 4


Add.

2 – 4 + 8 – 3 + –5 – 4 + 2 2x x x x

= – 4x + 8x2 – 3 – 5x2 – 4 + 2x

= ( 8x2 – 5x2) + ( – 4x + 2x) + ( – 3 – 4)

= 3x2 – 2x – 7

Section 4.5

Exercise #12

Chapter 4


23 – 2 – 5 – 2 + 82x x x x

= 3x2 – 2x – 5x + 2 – 8x2

= (3x2 – 8x2) + ( – 2x – 5x ) +2

= – 5x2 – 7x +2

Subtract 5x – 2 + 8x2 from 3x2 – 2x.

Section 4.6

Multiplication

of Polynomials

OBJECTIVES

Multiply two monomials. A

Multiply a monomial and

a binomial.

B

OBJECTIVES

Multiply two binomials

using FOIL method.

C

Solve an application. D

PROCEDURE

First terms multiplied first.

FOIL Method for Multiplying

Binomials

Outer terms multiplied second.

Inner terms multiplied third.

Last terms multiplied last.

Section 4.6

Exercise #16

Chapter 4


Find (5x – 2y) (4x – 3y) .

= 20x2 – 23xy + 6y2

= 20x2 – 15xy – 8xy + 6y2F O I L

Section 4.7

Special Product

of Polynomials

OBJECTIVES

Expand binomials of the form

A (X +A)2

B (X – A)2

C (X +A)(X – A)

OBJECTIVES

Multiply a binomial by a

trinomial.

D

Multiply any two

polynomials.

E

SPECIAL PRODUCTS

(X +A)(X +B)= X 2+(A+B)X +AB

SP1 or FOIL

SPECIAL PRODUCTS

SP2

(X +A)(X +A)=(X +A)2

= X 2+2AX +A2

SPECIAL PRODUCTS

SP3

(X -A)(X -A)=(X -A)2

= X 2 -2AX +A2

SPECIAL PRODUCTS

2 2( + )( - )= -X A X A X A

SP4

PROCEDURE

Multiplying Any Two

Polynomials

(Term-By-Term Multiplication)

Multiply each term of one by

every term of other and add

results.

PROCEDURE

Appropriate Method for

Multiplying Two Polynomials:

1. Is the product the square

of a binomial?

Both answers have three terms.

If so, use SP2 or SP3.

PROCEDURE



2. Are the two binomials in the

product the sum and

difference of the same two

terms?

PROCEDURE



Answer has two terms.

If so, use SP4.

PROCEDURE



3. Is the binomial product

different from previous

two?

Answer has three or four terms. If so, use FOIL.

PROCEDURE



4. Is product still different?

If so, multiply every term

of first polynomial by

every term of second and

collect like terms.

Section 4.7

Exercise #18

Chapter 4


Expand.

(2x – 7y)2

(a – b)2

= a2– 2 ab + b

2

= 4x2 – 28xy + 49y2

= (2x)2– 2 (2x)(7y) + ( 7y ) 2

Section 4.7

Exercise #19

Chapter 4


Find (2x – 5y)(2x + 5y).

= (2x)2 – (5y)2

= 4x2 – 25y2

Section 4.7

Exercise #20

Chapter 4


Find (x + 2)(x2 + 5x + 3)

= x(x2 + 5x + 3) + 2(x2 + 5x + 3)

= x 3 + 5x2 + 3x + 2x2 + 10x + 6

= x 3 + (5x2 + 2x2 ) + (3x + 10x) + 6

= x 3 + 7x2 + 13x + 6

Section 4.8

Division

of Polynomials

OBJECTIVES

Divide a polynomial by a

monomial.

A

Divide one polynomial

by another polynomial.

B

RULE

To Divide A Polynomial By A

Monomial

Divide each term in

polynomial by monomial.

Section 4.8

Exercise #25

Chapter 4


x – 2 2x3 + 0x 2 – 9x + 5

2x3 – 4x 2

4x 2 – 9x + 5

4x 2 – 8x

– 1x + 5

– 1x + 2

3

2x 2 + 4x – 1 R 3

Divide.

2x3 – 9x + 5 by x – 2

Remainder

the product, quotient, and power rules for exponents...quotient rule for exponents example: p 8 p3 =...

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