section 4.1 the product, quotient, and power rules for exponents

Section 4.1

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

RULESSigns for Multiplication

1. When multiplying two numbers with the same sign, product is positive (+).

RULESSigns for Multiplication

2. When multiplying two numbers with different signs, product is negative (-).

RULESSigns for Division

1.When dividing two numbers with the same sign, product is positive (+).

RULESSigns for Division

2.When dividing two numbers with different signs, product is negative (-).

RULES FOR EXPONENTSIf m, n, and k are positive integers, then:1. Product rule for exponents

xmxn = xm+n

Example:

x5•x6 = x5+6 = x11

RULES FOR EXPONENTSIf m, n, and k are positive integers, then:2. Quotient rule for exponents

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

x

RULES FOR EXPONENTSIf m, n, and k are positive integers, then:2. Quotient rule for exponents

Example:

p8

p3 = p8-3 = p5

RULES FOR EXPONENTSIf m, n, and k are positive integers, then:

3. Power rule for products

=k mk nkm n yy xx

RULES FOR EXPONENTSIf m, n, and k are positive integers, then:

3. Power rule for products

Example:

= =4

4 3 4 4 3 4 16 12x y x xy y• •

RULES FOR EXPONENTSIf m, n, and k are positive integers, then:

4. Power rule for quotients

0=m m

m yx x y y

RULES FOR EXPONENTSIf m, n, and k are positive integers, then:

4. Power rule for quotients Example:

6

= =3 3 6 184 4 6 24

a a ab b b

••

Section 4.1Exercise #1

Chapter 4Exponents and Polynomials

Find.

a. (2a3b)(– 6ab3 )

= (2 • – 6)a3+1 b1+3

= – 12a4b4

b. (– 2x 2yz)(– 6xy3z 4)

= ( – 2 • – 6)x 2 + 1 y1 + 3 z1 + 4

= 12x3y5z5

Find.

c. 18x5y7

– 9xy3

= 18

– 9

x5 – 1 y7 – 3

= – 2x 4y 4

Section 4.1Exercise #2

Chapter 4Exponents and Polynomials

Find.

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

= 8x 9y6

b. ( – 3x 2y3 )2

a. (2x3y 2 )3

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

= 9x 4y6

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

RULESZero Exponent

0For 0, =1x x

– n 1= 0nx xx

If n is a positive integer,Negative Exponent

RULESnth Power of a Quotient

–1 =

nnxx

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 4.2Exercise #4

Chapter 4Exponents and Polynomials

Simplify and write the answer without negative exponents.

– 71a. 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 4.2Exercise #5

Chapter 4Exponents and Polynomials

Simplify.– 3 4

2 3

– 2

2

b. 3

x yx 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

2 x

y

= 9x10

4y2

Section 4.3

Applicationof 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)Writing a number in scientific notation

If the decimal point is moved to the right, n is negative.

PROCEDURE

3. Write (M10n).

(M10n)Writing a number in scientific notation

PROCEDUREMultiplying using scientific notation

1. Multiply decimal parts first. Write result in scientific notation.

PROCEDUREMultiplying using scientific notation

2. Multiply powers of 10 using product rule.

PROCEDUREMultiplying using scientific notation

3. Answer is product obtained in steps 1 and 2 after simplification.

Section 4.3Exercise #6

Chapter 4Exponents and Polynomials

a. 48,000,000

Write in scientific notation.

= 4 8000000 .

= 4.8107

b. 0.00000037

= 0.0000003 7

= 3.7 10 – 7

Section 4.3Exercise #7

Chapter 4Exponents and Polynomials

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

DEFINITIONPolynomialAn algebraic expression formed using addition and subtraction on products of numbers and variables raised to whole number exponents.

Section 4.4Exercise #8

Chapter 4Exponents and Polynomials

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

a. 3x – 5

B, binomial

b. 5x3

M, monomial

c. 8x 2 – 2 + 5x

T, trinomial

Section 4.4Exercise #10

Chapter 4Exponents and Polynomials

Find the value.

– 16t 2 + 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 4.5Exercise #11

Chapter 4Exponents and Polynomials

Add.

2 – 4 + 8 – 3 + –5 – 4 + 2 2x x x x

= – 4x + 8x 2 – 3 – 5x 2 – 4 + 2x

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

= 3x 2 – 2x – 7

Section 4.5Exercise #12

Chapter 4Exponents and Polynomials

23 – 2 – 5 – 2 + 82x x x x

= 3x 2 – 2x – 5x + 2 – 8x 2

= (3x 2 – 8x 2) + ( – 2x – 5x ) + 2

= – 5x 2 – 7x +2

Subtract 5x – 2 + 8x 2 from 3x2 – 2x.

Section 4.6

Multiplicationof 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 4.6Exercise #16

Chapter 4Exponents and Polynomials

Find (5x – 2y ) (4x – 3y ) .

= 20x 2 – 23xy + 6y 2

= 20x 2 – 15xy – 8xy + 6y 2F O I L

Section 4.7

Special Productof 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

PROCEDUREMultiplying Any Two Polynomials (Term-By-Term Multiplication)

Multiply each term of one by every term of other and add results.

PROCEDUREAppropriate 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.

PROCEDUREAppropriate Method for Multiplying Two Polynomials:2. Are the two binomials in the

product the sum and difference of the same two terms?

PROCEDUREAppropriate Method for Multiplying Two Polynomials:

Answer has two terms.

If so, use SP4.

PROCEDUREAppropriate Method for Multiplying Two Polynomials:

3. Is the binomial product different from previous two?

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

PROCEDUREAppropriate Method for Multiplying Two Polynomials:

4. Is product still different? If so, multiply every term of first polynomial by every term of second and collect like terms.

Section 4.7Exercise #18

Chapter 4Exponents and Polynomials

Expand.

(2x – 7y )2 (a – b)

2 = a

2– 2 ab + b

2

= 4x 2 – 28xy + 49y 2

= (2x)2

– 2 (2x)(7y) + ( 7y ) 2

Section 4.7Exercise #19

Chapter 4Exponents and Polynomials

Find (2x – 5y )(2x + 5y).

= (2x )2 – (5y )2

= 4x 2 – 25y 2

Section 4.7Exercise #20

Chapter 4Exponents and Polynomials

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

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

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

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

= x 3 + 7x 2 + 13x + 6

Section 4.8

Divisionof Polynomials

OBJECTIVES

Divide a polynomial by a monomial.

A

Divide one polynomial by another polynomial.

B

RULETo Divide A Polynomial By A Monomial

Divide each term in polynomial by monomial.

Section 4.8Exercise #25

Chapter 4Exponents and Polynomials

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