2. Warm Up
Evaluate each expression for the given values of
the variables.
1. x3y2 for x = β1 and y = 10 β100
2. for x = 4 and y = (β7)
Write each number as a power of the given base.
3. 64; base 4 43
4. β27; base (β3) (β3)3
3. You have seen positive exponents. Recall that to
simplify 32, 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.
Power 55 54 53 52 51 50 5β1 5β2
3125 625 125 25 5
Value
5 5 5 5
4. When the exponent decreases by one, the value of
the power is divided by 5. Continue the pattern of
dividing by 5.
7. Notice the phrase βnonzero numberβ 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 0ΒΊ = . Also 0β6 would be = . Since division
by 0 is undefined, neither value exists.
10. Check It Out! Example 1
A sand fly may have a wingspan up to 5β3 m. Simplify this
expression.
5-3 m is equal to
11. Example 2: Zero and Negative Exponents
Simplify.
A. 4β3
B. 70 Any nonzero number raised to the zero power is 1.
7ΒΊ = 1
C. (β5)β4
D. β5β4
12. Caution
In (β3)β4, the base is negative because the negative sign is inside the
parentheses. In β3β4 the base (3) is positive.
13. Check It Out! Example 2
Simplify.
a. 10β4
b. (β2)β4
c. (β2)β5
d. β2β5
14. Example 3A: Evaluating Expressions with Zero and Negative Exponents
Evaluate the expression for the given value of the variables.
xβ2 for x = 4
Substitute 4 for x.
Use the definition
15. Example 3B: Evaluating Expressions with Zero and Negative Exponents
Evaluate the expression for the given values of the variables.
β2a0b-4 for a = 5 and b = β3
Substitute 5 for a and β3 for b.
Evaluate expressions with exponents.
Write the power in the denominator as a
product.
Evaluate the powers in the
product.
Simplify.
16. Check It Out! Example 3a
Evaluate the expression for the given value of the variable.
pβ3 for p = 4
Substitute 4 for p.
Evaluate exponent.
Write the power in the denominator as a
product.
Evaluate the powers in the
product.
17. Check It Out! Example 3b
Evaluate the expression for the given values of the variables.
for a = β2 and b = 6
Substitute β2 for a and 6 for b.
Evaluate expressions with exponents.
Write the power in the denominator as a
product.
Evaluate the powers in the
product.
Simplify.
2
18. What if you have an expression with a negative exponent in a denominator, such as ?
or Definition of a negative
exponent.
Substitute β8 for n.
Simplify the exponent on the
right side.
So ifexpression that contains negativeis in a denominator, it is equivalent to the same
An a base with a negative exponent or zero exponents is not considered to be
base with the opposite (positive) exponent in the numerator. exponents.
simplified. Expressions should be rewritten with only positive
19. Example 4: Simplifying Expressions with Zero and Negative Numbers
Simplify.
A. 7wβ4 B.