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

Now let’s think about the other scenario. In this case, we have (-2)

Example 1: Evaluate: (-5)

(-5)

Example 2: Evaluate: -4

-4

Example 3: Evaluate: (-10)

(-10)

Example 4: Determine the sign only: (-29)

- -2
^{2}≠ (-2)^{2} - -2
^{2}» -1 x 2^{2}» -1 x 2 x 2 = -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 2 x 2 = -4Now 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 4 x 4 x 4 = -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: (-5)

^{3}(-5)

^{3}= -5 x -5 x -5 = -125Example 2: Evaluate: -4

^{2}-4

^{2}= -1 x 4 x 4 = -16Example 3: Evaluate: (-10)

^{4}(-10)

^{4}= -10 x -10 x -10 x -10 = 10,000Example 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 (-)

^{8}- Our base is 14, the negative is not wrapped inside of parentheses
- Our result will be negative (-)

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