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Exponents with Negative Bases



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In this section, we explore the topic of exponents with negative bases. When we perform operations with exponents and the base is a positive whole number, things are fairly straight forward.
43 = 4 • 4 • 4 = 64
We simply have 3 factors of 4
When we start involving negative numbers as the base, we run into some confusion. Let's suppose we see: -52
Does this problem break down into 2 factors of (-5)? No, it does not and this is where we have to pay close attention. Notice how -5 is not inside of parentheses. This tells us we could really write the problem as:
-52 = -1 • 52 = -1 • 25 = -25
The reason 5 is raised to the 2nd power first is due to the order of operations, exponents have a higher priority level than multiplication. Now let's suppose we saw:
(-5)2
This problem is telling us that we have 2 factors of (-5), the difference is that (-5) is enclosed in parentheses, so both the negative and the 5 are raised to the 2nd power:
(-5)2 = (-5)(-5) = 25
This shows us we must pay close attention to the details when working with exponents with negative bases. When the base does not have parentheses, we approach the problem differently.
Example 1: (-3)4
(-3)4 = (-3)(-3)(-3)(-3) = 81
Example 2: -72
-72 = -1 • 72 = -1 • 49 = -49
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