Lesson Objectives
 Demonstrate an understanding of operations with exponents
 Demonstrate an understanding of the order of operations
 Learn how to evaluate an exponential expression with a negative base
 Learn how to quickly determine the sign when working with exponent operations
How to Simplify an Exponential Expression with a Negative Base
When we work with exponents, we need to be extra cautious when dealing with negative numbers. If we are working with a negative number raised to a power, the base does not include the negative part unless we use parentheses:
We can really think about: 2^{2} as 1 x 2^{2}. From the order of operations, we know that we must perform exponent operations before we multiply. In this case, we would raise 2 to the 2nd power first, and then multiply the result by 1. This leads to 4 x 1, which gives us 4.
2^{2} = 1 x 2^{2} = 1 x 4 = 4
Now let’s think about the other scenario. In this case, we have (2)^{2}. Since the negative is wrapped inside of the parentheses, both are now part of the base. We can now show this as: (2)^{2} = 2 x 2 = 4. Let's think about another scenario:
Example 1: Evaluate each.
(5)^{3}
(5)^{3} = 5 x 5 x 5 = 125
Example 2: Evaluate each.
4^{2}
4^{2} = 1 x (4 x 4) = 16
Example 3: Evaluate each.
(10)^{4}
(10)^{4} = 10 x 10 x 10 x 10 = 10,000
Example 4: Determine the sign only.
(29)^{7}
14^{8}
 2^{2} ≠ (2)^{2}
 2^{2} » 1 x 2^{2} » 1 x 4 = 4
 (2)^{2} » 2 x 2 = 4
We can really think about: 2^{2} as 1 x 2^{2}. From the order of operations, we know that we must perform exponent operations before we multiply. In this case, we would raise 2 to the 2nd power first, and then multiply the result by 1. This leads to 4 x 1, which gives us 4.
2^{2} = 1 x 2^{2} = 1 x 4 = 4
Now let’s think about the other scenario. In this case, we have (2)^{2}. Since the negative is wrapped inside of the parentheses, both are now part of the base. We can now show this as: (2)^{2} = 2 x 2 = 4. Let's think about another scenario:
 4^{3} = 1 x 4^{3} = 1 x 64 = 64
 (4)^{3} = 4 x 4 x 4 = 64
Sign rules for Evaluating an Exponent with a Negative Base
 When the base is (), and enclosed inside of parentheses:
 The result is (+) if the exponent is even
 The result is () if the exponent is odd
 When the base is (), and not enclosed inside of parentheses:
 The result is always ()
Example 1: Evaluate each.
(5)^{3}
(5)^{3} = 5 x 5 x 5 = 125
Example 2: Evaluate each.
4^{2}
4^{2} = 1 x (4 x 4) = 16
Example 3: Evaluate each.
(10)^{4}
(10)^{4} = 10 x 10 x 10 x 10 = 10,000
Example 4: Determine the sign only.
(29)^{7}
 Our base 29 is wrapped inside of parentheses
 The exponent 7, is an odd number
 Our result will be negative ()
14^{8}
 Our base is 14, the negative is not wrapped inside of parentheses
 Our result will be negative ()
Skills Check:
Example #1
Evaluate each. $$(13)^2$$
Please choose the best answer.
A
169
B
169
C
26
D
26
E
132
Example #2
Evaluate each. $$7^{4}$$
Please choose the best answer.
A
2401
B
2401
C
74
D
74
E
11
Example #3
Determine the sign. $$(59)^{33}$$
Please choose the best answer.
A
+
B

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