6-1 Integer
Exponents
Property
Product
of Power of a
Powers
Power
Power of a
Product
Words
To multiple
powers that
have the
same base,
add the
exponents
To find a
power of a
power, multiply
the exponents
To find a
power of a
product, find
the power of
each factor
and multiply
the exponents
Algebra
am•an=am+n
(am)n=amn
(ab)m=ambm
Example 56•59=56+9 (34)2=34•2=38 (232•17)5=
10175
15
23
=5
Holt McDougal Algebra 1
6-1 Integer Exponents
Property
Quotient of
Power of a
Powers
Quotient
Words
Algebra
Example
Holt McDougal Algebra 1
To divide
powers having
the same base,
subtract the
exponents
m
a
mn
a
n
a
7
4
72
4
2
4
To find a power of
a quotient, find the
power of the
numerator and the
denominator and
divide
m
a
a
   m
b 7 b
7
3
3
   7
2
2
m
6-1 Integer Exponents
Holt McDougal Algebra 1
6-1 Integer Exponents
Notice the phrase “nonzero number” in the
previous table. This is because 00 and 0 raised to
a negative power are both undefined. For
example, if you use the pattern given above the
table with a base of 0 instead of 5, you would
get 0º =
. Also 0–6 would be
=
. Since
division by 0 is undefined, neither value exists.
Holt McDougal Algebra 1
6-1 Integer Exponents
Example 2: Zero and Negative Exponents
Simplify.
A. 4–3
B. 70
7º = 1
C. (–5)–4
D. –5–4
Holt McDougal Algebra 1
6-1 Integer Exponents
Caution
In (–3)–4, the base is negative because the
negative sign is inside the parentheses. In –3–4
the base (3) is positive.
Holt McDougal Algebra 1
6-1 Integer Exponents
Simplify.
a. 10–4
Check It Out! Example 2
b. (–2)–4
c. (–2)–5
d. –2–5
Holt McDougal Algebra 1
6-1 Integer Exponents
Example 3A: Evaluating Expressions with Zero and
Negative Exponents
Evaluate the expressions for the given value of
the variables.
x–2 for x = 4
p–3 for p = 4
–2a0b-4 for a = 5 and b = –3
for a = –2 and b = 6
Holt McDougal Algebra 1
2
6-1 Integer Exponents
What if you have an expression with a negative
exponent in a denominator, such as
?
***An
expression
that contains
negative
So if a base
with a negative
exponent
is in a or zero
exponents
isitnot
considered
to be
simplified.
denominator,
is equivalent
to the
same
base with
Expressions
should be
rewritten
with
only
the opposite (positive)
exponent
in the
numerator.
positive exponents.***
Holt McDougal Algebra 1
6-1 Integer Exponents
Simplify the expression.
5.
(42)7 = 414
6.
[(–2)4]5
7.
(n3)6 = n18
8.
[(m + 1)5]4 = (m + 1)20
Holt McDougal Algebra 1
= (–2)20
Example 2 Exponents
6-1 Integer
Simplify the expression.
x 3 x3
a.
= 3
y
y
b.
–7
x
2
=
–7
x
Holt McDougal Algebra 1
2
(– 7)2
49
=
=
x2
x2
6-1 Integer Exponents
Find the values of x and y with the given information.
y
b
3

b
x
b
y
2
b b
13

b
3x
b
Holt McDougal Algebra 1
6-1 Integer Exponents
Solve for x and/or y
1.
4 2
x
4 8  2
x
2.
64
82
Holt McDougal Algebra 1
3
y 1
y
6-1 Integer Exponents
Simplify each expression.
1.
2.
4
1
10
3.
4.
–3
Holt McDougal Algebra 1
6-1 Integer Exponents
Another way to write nth roots is by using fractional
exponents. For example, for b >1, suppose
b1 = b2k
1 = 2k
Square both sides.
Power of a Power Property
If bm = bn, then m = n.
Divide both sides by 2.
So for all b > 1,
Holt McDougal Algebra 1
6-1 Integer Exponents
Check It Out! Example 1
Simplify each expression.
a.
Use the definition of
1
b .n
=3
b.
Use the definition of
= 11 + 4
= 15
Holt McDougal Algebra 1
1
b .n
6-1 Integer Exponents
Additional Example 2: Simplifying Expressions with
Fractional Exponents
Simplify each expression.
A.
B.
= 243
C.
= 25
D.
=8
E.
= 81
Holt McDougal Algebra 1
=1
6-1 Integer Exponents
Additional Example 4B: Properties of Exponents to
Simplify Expressions
Simplify. All variables represent nonnegative
numbers.
Holt McDougal Algebra 1
6-1 Integer Exponents
Holt McDougal Algebra 1
6-1 Integer Exponents
A monomial is a number, a variable, or a product of
numbers and variables with whole-number
exponents.
Holt McDougal Algebra 1
6-1 Integer Exponents
Monomials
NOT a monomial
5+z
2/n
4a
x-1
Holt McDougal Algebra 1
Reason
A sum is not a monomial
A monomial cannot
have a variable
denominator
A monomial cannot
have a variable
exponent
The variable must have
a whole number
exponent.
6-1 Integer Exponents
Monomial
10
3x
Degree
0
1
1+2=3
-1.8m5
5
The degree of a monomial is the sum of the
exponents of the variables. A constant has
degree 0.
Holt McDougal Algebra 1
6-1 Integer Exponents
Example 1: Finding the Degree of a Monomial
Find the degree of each monomial.
A. 4p4q3
The degree is 7.
B. 7ed
The degree is 2.
C. 3
The degree is 0.
Holt McDougal Algebra 1
6-1 Integer Exponents
Check It Out! Example 1
Find the degree of each monomial.
a. 1.5k2m
The degree is 3.
b. 4x
The degree is 1.
c. 2c3
The degree is 3.
Holt McDougal Algebra 1
6-1 Integer Exponents
A polynomial is a monomial or a sum or
difference of monomials.
Each monomial in a polynomial is called a
term.
The degree of a polynomial is the degree
of the term with the greatest degree.
Holt McDougal Algebra 1
6-1 Integer Exponents
Polynomials
Degree of
polynomial
2 x  x  5 x  12
3
Leading
Coefficient
Holt McDougal Algebra 1
2
Constant
term
6-1 Integer Exponents
Special Polynomials
• Binomial
– Polynomial with two terms
• Trinomial
– Polynomial with three terms
Holt McDougal Algebra 1
6-1 Integer Exponents
Example 2: Finding the Degree of a Polynomial
Find the degree of each polynomial.
A. 11x7 + 3x3
The degree of the polynomial is
the greatest degree, 7.
B.
The degree of the polynomial is the greatest degree, 4.
Holt McDougal Algebra 1
6-1 Integer Exponents
Check It Out! Example 2
Find the degree of each polynomial.
a. 5x – 6
The degree of the polynomial
is the greatest degree, 1.
b. x3y2 + x2y3 – x4 + 2
The degree of the polynomial is
the greatest degree, 5.
Holt McDougal Algebra 1
6-1 Integer Exponents
Holt McDougal Algebra 1
6-1 Integer Exponents
Some polynomials have special names based on
their degree and the number of terms they have.
Degree
Name
Terms
Name
0
Constant
1
Monomial
1
Linear
2
Binomial
2
Quadratic
Trinomial
3
4
Cubic
Quartic
3
4 or
more
5
Quintic
6 or more
Holt McDougal Algebra 1
6th,7th,degree
and so on
Polynomial
6-1 Integer Exponents
Example 4: Classifying Polynomials
Classify each polynomial according to its degree
and number of terms.
A. 5n3 + 4n
cubic binomial.
B. 4y6 – 5y3 + 2y – 9
6th-degree polynomial.
C. –2x
linear monomial.
Holt McDougal Algebra 1
6-1 Integer Exponents
Check It Out! Example 4
Classify each polynomial according to its degree
and number of terms.
a. x3 + x2 – x + 2
cubic polynomial.
b. 6
constant monomial.
c. –3y8 + 18y5 + 14y
8th-degree trinomial.
Holt McDougal Algebra 1
Example 2 Exponents
6-1 Integer
Tell whether is a polynomial. If it is a polynomial, find its
degree and classify it by the number of its terms.
Otherwise, tell why it is not a polynomial.
Expression
Is it a polynomial?
Classify by degree and
number of terms
a.
9
Yes
constant monomial
b.
c.
d.
e.
2x2 + x – 5
Yes
Quadratic trinomial
6n4 – 8n
No; variable exponent
n– 2 – 3
No; negative exponent
7bc3 + 4b4c
Yes
Holt McDougal Algebra 1
Quintic binomial
6-1 Integer Exponents
Example 5: Application Continued
A tourist accidentally drops her lip balm off the
Golden Gate Bridge. The bridge is 220 feet from the
water of the bay. The height of the lip balm is given
by the polynomial –16t2 + 220, where t is time in
seconds. How far above the water will the lip balm be
after 3 seconds?
After 3 seconds the lip balm will be 76 feet
from the water.
Holt McDougal Algebra 1
6-1 Integer Exponents
Check It Out! Example 5
What if…? Another firework with a 5-second fuse is
launched from the same platform at a speed of 400
feet per second. Its height is given by –16t2 +400t
+ 6. How high will this firework be when it
explodes?
1606 feet
Holt McDougal Algebra 1
6-1 Integer Exponents
Solve for x and/or y
1.
4 2
x
4 8  2
x
2.
64
82
Holt McDougal Algebra 1
3
y 1
y
6-1 Integer Exponents
Simplify the expression. Write your answer using only
positive exponents.
a.
b.
8x3
= 15
y
(2xy–5)3
(2x)–2y5
–4x2y2
8
= –
12 x y
4x
2
y
7

6 2
Holt McDougal Algebra 1
y3
16x4
6-1 Integer Exponents
Lesson Quiz: Part II
5. In an experiment, the approximate population P
of a bacteria colony is given by
, where t is the number of days since
start of the experiment. Find the population of the
colony on the 8th day.
480
Simplify. All variables represent nonnegative
numbers.
6.
7.
Holt McDougal Algebra 1
6-1 Integer Exponents
Lesson Quiz: Part I
Find the degree of each polynomial.
1. 7a3b2 – 2a4 + 4b – 15
2. 25x2 – 3x4
5
4
Write each polynomial in standard form. Then give
the leading coefficient.
3. 24g3 + 10 + 7g5 – g2
4. 14 – x4 + 3x2
Holt McDougal Algebra 1
7g5 + 24g3 – g2 + 10; 7
–x4 + 3x2 + 14; –1
6-1 Integer Exponents
Lesson Quiz: Part II
Classify each polynomial according to its degree
and number of terms.
5. 18x2 – 12x + 5
6. 2x4 – 1
quadratic trinomial
quartic binomial
7. The polynomial 3.675v + 0.096v2 is used to
estimate the stopping distance in feet for a car
whose speed is v miles per hour on flat dry
pavement. What is the stopping distance for a
car traveling at 70 miles per hour?
727.65 ft
Holt McDougal Algebra 1