Exponents on negative bases
When the base is actually negative and when it isn’t
There are two cases to think of when you’re simplifying powers of negative bases.
Case 1: A negative sign directly in front of the base, ???b???, raised to a power.
If you have something that looks like this:
???-b^a???
where ???a??? and ???b??? are both positive real numbers, it’s really the same as ???-1*b^a???. This is the case to watch out for because many people don’t realize when they see something like, ???-4^2???, it means the same thing as, ???-1(4^2)??? or ???-1(4\cdot4)=-16???.
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This is because PEMDAS and the order of operations tells us that you need to do the exponent first and then multiply by the negative sign.
Case 2: A negative sign included in the parenthesis.
If you have something that looks like this: ???(-b)^a???, where ???a??? and ???b??? are both positive real numbers then raise everything inside the parenthesis to the power of ???a???, in other words multiply ???-b??? by itself ???a??? times, it’s like having ???a??? factors of the number ???-b???.
This is the case most people think of when they perform operations with exponents. This means ???(-4)^2??? equals ???(-4)(-4)??? or ???16???.
How to apply an exponent to a negative base
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Finding powers of negative bases
Example
Simplify the expression.
???-2^3???
This is an example of Case 1. PEMDAS and the order of operations tells us that we should do the exponent first, and then multiply by the negative sign. Remember the negative sign is the same as multiplying by ???-1???.
???-2^3???
???-(2\cdot2\cdot2)???
???-(8)???
???-8???
Let’s take a look at another example with a negative number inside parentheses.
This is because PEMDAS and the order of operations tells us that you need to do the exponent first and then multiply by the negative sign.
Example
Simplify the expression.
???(-1)^4???
Remember that ???-1^4??? is different than ???(-1)^4???. When we have ???(-1)^4??? the negative sign is included in the parenthesis. This means we need to raise everything in the parenthesis to the power of ???4???, so it’s like having four factors of ???-1???.
???(-1)^4???
???(-1)(-1)(-1)(-1)???
???1???
