Warm Up
Evaluate each expression for the given values of the variables.
1.
x3y2 for x = –1 and y = 10
2.
3𝑥 2
𝑦2
for x = 4 and y = (–7)
Write each number as a power of the given base.
3.
64; base 4
4.
–27; base (–3)
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Rules for Exponents
Product of Powers
Power of a Power
Power of a Product
Power of a
Monomial
Quotient of Powers
Rule
For any number a, and all integers m and n,
𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛
For any number a, and all integers m and n,
(𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
For all numbers a and b, and any integer m,
(𝑎𝑏)𝑚 = 𝑎𝑚 𝑏 𝑛
For all numbers a and b, and all integer m, n, and
p,
(𝑎𝑚 𝑏 𝑛 )𝑝 = 𝑎𝑚𝑝 𝑏 𝑛𝑝
For all integers m and n, and any nonzero number
a,
𝑎𝑚
= 𝑎𝑚−𝑛
𝑎𝑛
Example
𝑎2 ∙ 𝑎6 = 𝑎2+6 = 𝑎8
(𝑥 2 )7 = 𝑥 2∙7 = 𝑥 14
(𝑝𝑞)4 = 𝑝4 𝑞 4
(𝑤 4 𝑡)3 = (𝑤 4 𝑡 1 )3 = 𝑤 4∙3 𝑡 1∙3 = 𝑤 12 𝑡 3
𝑞9
= 𝑞 9−4 = 𝑞 5
𝑞4
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Algebra/Lesson 6-1: Integer Exponents
Objectives:


Evaluate expressions containing zero and integer exponents.
Simplify expressions containing zero and integer exponents.
1
You have seen positive exponents. Recall that to simplify 3 2, use 3 as a factor 2 times: 32 = 3  3 = 9.
But what does it mean for an exponent to be negative or 0? You can use a table and look for a pattern to figure it out.
When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5.
Notice the phrase “nonzero number” in the previous table. This is because 0 0 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 =
0
0
. Also 0–6 would be
1
06
1
= .
0
Since division by 0 is undefined, neither value exists.
Example 1:
One cup is 2–4 gallons. Simplify this expression.
C.I.O.-Example 1:
A sand fly may have a wingspan up to 5–3 m. Simplify this expression.
2
Example 2: Simplify.
A.
4–3
B.
70
C.
(–5)–4
D.
–5–4
D.
–2–5
C.I.O.-Example 2: Simplify.
A.
10–4
B.
(–2)–4
C.
(–2)–5
Example 3:
A. Evaluate the expression for the given value of the
variables:
x–2 for x = 4
B. Simplify the expression for the given values of the
variables:
–2a0b-4 for a = 5 and b = –3
C.I.O.-Example 3: Evaluate the expression for the given value of the variable.
a.
p–3 for p = 4
b.
8a–2b0 for a = –2 and b = 6
3
What if you have an expression with a negative exponent in a denominator, such as
1
𝑥 −𝑛
?
An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only
positive exponents.
Example 4: Simplify.
A.
7w–4
B.
−𝟓
𝒌−𝟐
C.
𝑎0 𝑏 −2
𝑐 −3 𝑑 6
C.I.O.-Example 4: Simplify.
a.
2r0m–3
b.
𝒓−𝟑
𝟕
c.
𝒈𝟒
𝒉−𝟔
Lesson Quiz: Part I
1. A square foot is 3–2 square yards. Simplify this expression.
4
Simplify.
2.
2–6
3.
(–7)–3
4.
60
5.
–112
Lesson Quiz: Part II
Evaluate each expression for the given value(s) of the variables(s).
6.
x–4 for x =10.
7.
2a–1b–3 for a = 6 and b = 3.
p. 395: 27-75 odd, 78-83
78) always
80) sometimes
82) never
5