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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
If m, n, and k are positive
integers, then:
2. Quotient rule for exponents
- > , 0=m m nn m n xx x
x
RULES FOR EXPONENTS
If m, n, and k are positive
integers, then:
2. Quotient rule for exponents
Example:
p8
p3= p8-3 = p5
RULES FOR EXPONENTS
If m, n, and k are positive
integers, then:
3. Power rule for products
=k
mk nkm n yy xx
RULES FOR EXPONENTS
If 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 EXPONENTS
If m, n, and k are positive
integers, then:
4. Power rule for quotients
0=m m
myx x
y y
RULES FOR EXPONENTS
If m, n, and k are positive
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
Exponents and Polynomials
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
Exponents and Polynomials
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
Exponents and Polynomials
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)
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
PROCEDURE
Multiplying using scientific
notation
1. Multiply decimal parts first.
Write result in scientific
notation.
PROCEDURE
Multiplying using scientific
notation
2. Multiply powers of 10
using product rule.
PROCEDURE
Multiplying using scientific
notation
3. Answer is product
obtained in steps 1 and 2
after simplification.
Section 4.3
Exercise #6
Chapter 4
Exponents 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 4.3
Exercise #7
Chapter 4
Exponents 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
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
Exponents and Polynomials
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
Exponents and Polynomials
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
Exponents and Polynomials
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
Exponents and Polynomials
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
Exponents and Polynomials
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
Appropriate Method for
Multiplying Two Polynomials:
2. Are the two binomials in the
product the sum and
difference of the same two
terms?
PROCEDURE
Appropriate Method for
Multiplying Two Polynomials:
Answer has two terms.
If so, use SP4.
PROCEDURE
Appropriate Method for
Multiplying Two Polynomials:
3. Is the binomial product
different from previous
two?
Answer has three or four terms. If so, use FOIL.
PROCEDURE
Appropriate 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 4.7
Exercise #18
Chapter 4
Exponents and Polynomials
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
Exponents and Polynomials
Find (2x – 5y)(2x + 5y).
= (2x)2 – (5y)2
= 4x2 – 25y2
Section 4.7
Exercise #20
Chapter 4
Exponents and Polynomials
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
Exponents 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