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Chapter 4, 5, and 13
Notes
4-7 Exponents and
Multiplication
 To multiply numbers or
variables with the same base,
add the exponents.
 Examples:
1. 3 • 32 = 3 • 3 • 3 = 31+2 = 33
2. 22 • 23 = 2 • 2 • 2 • 2 • 2 = 22+3
= 25
You can also do the same
thing with variables
a5 • a1 • b2
m5 • m7
y2 • y3 • z • z4
You can also do the same thing
with variables - answers
a5 • a1 • b2 = a5+1b2 = a6b2
m5 • m7 = m5+7 = m12
y2 • y3 • z • z4 = y2+3z1+4 = y5z5
Simplify:
1. -2x2 • 3x5
2. 6a3 • 3a
3. -5c2 • -3c7
Simplify: Answers
1. -2x2 • 3x5 = -2 • 3 • x2 • x5 = -6x7
2. 6a3 • 3a = 18a4
3. -5c2 • -3c7 = 15c9
Finding a Power of a Power
 To find a power of a power,
multiply the exponents

1.
2.
3.
Examples:
(32)3
(24)2
(72)3
Finding a Power of a Power
- Answers
 To find a power of a power,
multiply the exponents

1.
2.
3.
Examples:
(32)3 = (3)2 •3 = 36 = 729
(24)2 = (2)4 •2 = 28 = 256
(72)3 = (7)2 •3 = 76
 Simplify each expression:
1. (c5)4
2. (m3)2
3. (3x2)2
 Simplify each expression: Answers
1. (c5)4 = (c)5 •4 = c20
2. (m3)2 = (m)3 •2 = m6
3. (3x2)2 = 9x4
4-8 Exponents and Division
 To divide numbers or variables
with the same base, subtract the
exponents
 x0 = 1
 Examples:
1. 38
35
2. 107
104
4-8 Exponents and Division Answers

To divide numbers or variables with the
same base, subtract the exponents
 x0 = 1
 Examples:
1. 38 = 38-5 = 33 = 27
35
2. 107 = 107-4 = 103 = 1,000
104
Simplify each expression
1. X25
x18
2. 12m5
3m
Simplify each expression
1. x25
x18
= x25-18 = x7
2. 12m5 = 4m5-1 = 4m4
3m
Zero as an Exponent
1. (-8)2
(-8)2
2. 52x6
5x6
Zero as an Exponent Answers
1. (-8)2 = (-8)2-2 = (-8)0 = 1
(-8)2
2. 52x6 = 25x6 = 5x6-6 = 5
5x6
5x6
Negative/Positive
Exponents
1. 56
58
2. 3y8
9y12
Negative/Positive
Exponents - Answers
1. 56 = 56-8 = 5-2 = 1 = 1
58
52 25
2. 3y8 = 1y8 = 1
9y12 3y12 3y4
Using exponents without a
fraction bar
1. x2y3
x3y
2. m3n2
m6n8
Using exponents without a
fraction bar - Answers
1. x2y3 = x2-3y3-1 = x-1y2
x3y
2. m3n2 = m3-6n2-8 = m-3n-6
m6n8
5-9 Powers of Products and
Quotients
 To raise a product to a
power, raise each factor to
the power.
Examples:
1. (4x2)3
2. (xy2)5
5-9 Powers of Products and
Quotients - Answers
 To raise a product to a
power, raise each factor to
the power.
Examples:
1. (4x2)3 = (4)3(x2)3 = 64x6
2. (xy2)5 = x5y10
Working with a negative
sign
1.
2.
3.
4.
(-5x)2
-(5x)2
(-5a2b3)2
-(2y)4
Working with a negative
sign - Answers
1. (-5x)2 = (-5)2x2 = 25x2
2. -(5x)2 = -(52)(x2) = -25x2
3. (-5a2b3)2 = 25a4b3
4. -(2y)4 = -16y4
Powers of Quotients
 To raise a quotient to a
power, raise both the
numerator and denominator
to the power.
1. (3/b)2
2. (2x2/3)3
Powers of Quotients Answers
 To raise a quotient to a
power, raise both the
numerator and denominator
to the power.
1. (3/b)2 = 32/b2 = 9/b2
2. (2x2/3)3 = 8x6/27
13-4 Polynomials
Polynomial - a monomial or a
sum or difference of
monomials
Monomial - a real number, a
variable, or a product of a real
number and variables
Is the expression a
monomial? Explain
7x2y
8 + a
a/7y
5x/4
Is the expression a monomial?
Explain -Answers
 7x2y - yes, it is the product of 7 and the
variables x and y
 8 + a - no, the expression is a sum
 a/7y - no, the denominator contains a
variable
 5x/4 - yes, it is the product of 5/4 and
the variable x
Polynomial
# of terms
Examples
Monomial
1
4, 32, x, x2
Binomial
2
Trinomial
3
x-3, 5x+1,
x3-x
x2 + x + 1
x4 -2x -5
Naming a Polynomial
x - y
8xyz
y2 + 8y + 18
10
Naming a Polynomial Answers
x - y
----- binomial
8xyz ----- monomial
y2 + 8y + 18 ----- trinomial
10
------ monomial
Evaluating a Polynomial
 Substitute values for the
variables
m = 8 and p = -3
1. 2mp
2. 3m - 2p
3. m2 + 2p - 3
Evaluating a Polynomial Answers
 Substitute values for the
variables
m = 8 and p = -3
1. 2mp = -48
2. 3m - 2p = 30
3. m2 + 2p - 3 = 55
13-5 Add/Subtract
Polynomials
Add by combining like terms or
aligning like terms
Example:
(2x2 + 3x -1) + (x2 + x -3)=
(2x2 + x2) + (3x + x ) + (-1 + -3) =
3x2 + 4x -4
More Examples
1. (z2 + 5z + 4) + (2z2 -5)
2. (4x + 9y)
+(3x - 5y)
3. a2 + 6a - 4
+ 8a2 - 8a
More Examples - Answers
1.
2.
3.
(z2 + 5z + 4) + (2z2 -5) = 3z2 + 5z -1
(4x + 9y)
+(3x - 5y)
7x + 4y
a2 + 6a - 4
+ 8a2 - 8a
9a2 - 2a - 4
Subtracting Polynomials
To subtract polynomials, you have
to add the opposite of each term
in the 2nd polynomial.
Example:
(5x2 + 10x) - (3x - 12)
5x2 + 10x
+
- 3x + 12
5x2 + 7x + 12
More examples
1. (7a2 - 2a) - (5a2 + 3a)
2. (10z2 + 6z + 5) - (z2 -8z + 7)
3. (3w2 + 8 + v) - (5w2 -3 - 7v)
More examples - Answers
1. (7a2 - 2a) - (5a2 + 3a) =
2a2 - 5a
2. (10z2 + 6z + 5) - (z2 -8z + 7) =
9z2 + 14z - 2
3. (3w2 + 8 + v) - (5w2 -3 - 7v) =
-2w2 + 11 + 8v
13-6 Multiplying a
Polynomial by a Monomial
Use the distributive property to
multiply each term of the
polynomial by the monomial.
Example:
2x(x + 4) =
2x(x) + 2x(4) =
2x2 + 8x
More examples
1. 3x2(8x2 - 5x + 2)
2. 2b(b2 + 3b -6)
3. 3x (2x2 + x + 3)
4. x(x2 + 2x + 4)
More examples - Answers
1. 3x2(8x2 - 5x + 2) = 24x4 - 15x3 + 6x2
2. 2b(b2 + 3b -6) = 2b3 + 6b2 - 12b
3. 3x (2x2 + x + 3) = 6x3 + 3x2 + 9x
4. x(x2 + 2x + 4) = x3 + 2x2 + 4x