Exponents with Negative Bases Lesson

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:

• -2² ≠ (-2)²
• -2² » -1 x 2² » -1 x 4 = -4
• (-2)² » -2 x -2 = 4

We can see from the above example that parentheses around a negative base do make a difference. It won’t give us a different answer in every scenario, but it’s important to know what’s causing a different answer. Let’s break each case down step by step:
We can really think about: -2² as -1 x 2². 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.
-2² = -1 x 2² = -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:

• -4³ = -1 x 4³ = -1 x 64 = -64
• (-4)³ = -4 x -4 x -4 = -64

If we work through the example above, we see that we get the same answer whether or not we use parentheses around the base. The reason here is that our exponent (3) is odd. Recall that when multiplying with negative numbers, each pair of negatives yields a positive product. Therefore an even (2,4,6,8,10,...) number of negative factors produces a positive product. When we get an odd number (1,3,5,7,9,...) of negative factors the opposite is true and we will get a negative product. Since a whole number exponent larger than 1 tells us the number of times to multiply the base by itself, it also tells us whether or not we will have a positive or negative result.

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 (-)

Let's take a look at some examples.
Example 1: Evaluate each.
(-5)³
(-5)³ = -5 x -5 x -5 = -125
Example 2: Evaluate each.
-4²
-4² = -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 (-)

Example 5: Determine the sign only.
-14⁸

• Our base is 14, the negative is not wrapped inside of parentheses
• Our result will be negative (-)

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