What happens when we multiply (−1) to itself multiple times? There is a pattern to find that makes simplifying exponent problems with a negative base much more simple.

Video Source (07:57 mins) | Transcript

Video Source (08:22 mins) | Transcript

The basic rule when a negative number is raised to an exponent \( (−b)^x \):

• If the power is even → the answer is positive.
• If the power is odd → the answer is negative.

It is important to know that \( −b^x \) is different than \( (−b)^x \). When you have \( −b^x \) , where the negative is not inside the parentheses, the exponent does not apply to it. This is because of the order of operations.

Additional Resources

• Khan Academy: Exponents with Negative Bases (04:29 mins, Transcript)

Practice Problems

1. \(({-}1)^{5}=?\) (
   Solution

   x

   Solution:
   \(-1\)
   Details:
   The rule is that \((−1)\) raised to an odd-numbered power is negative.
   
   Since 5 is an odd number, our answer is \(−1\).
   
   The following may help illustrate why this is true.
   
   \((-1)^{5}\) is the same as \((−1)\) multiplied together 5 times.
   
   
   
   Now let’s apply the following rules of multiplication:
   
   A negative times a negative is a positive number
   
   A negative times a positive is a negative number
   
   We can take the first and second \((−1)\) and multiply them together to get 1. We can do the same thing for the third and fourth \((−1)\).
   
   
   
   This leaves us with \((1)(1)(−1)\). Since 1 multiplied to anything is just that thing, all we have left is \((−1)\).
   
   
   )
2. \(({-}1)^{4}=?\) (
   Solution

   x

   Solution:
   1
   )
3. \(({-}1)^{105}=?\) (
   Solution

   x

   Solution:
   \(-1\)
   )
4. \(({-}1)^{236}=?\) (
   Solution

   x

   Solution:
   1
   )
5. \(({-}5)^{3}=?\) (
   Video Solution

   x

   Solution:
   \(-125\)
   Details:
   

   
   (Video Source | Transcript)
   )
6. \(({-}7)^{4}=?\) (
   Solution

   x

   Solution:
   2401
   )
7. \((-{\text{a}})^{6}=?\) (
   Video Solution

   x

   Solution:
   \({\text{a}}^{6}\)
   Details:
   

   
   (Video Source | Transcript)
   )
8. \(-1^{2}=?\) (
   Solution

   x

   Solution:
   \(-1\)
   Details:
   In our previous examples, the negative was inside the parentheses, so it was being raised to the exponent as well. If there aren’t parentheses, the exponent doesn’t apply to it. This is due to the order of operations.
   
   \(-1^{2} = -(1)(1) = -(1) = -1\)
   )
9. \(-2^{4}=?\) (
   Solution

   x

   Solution:
   \(-16\)
   Details:
   Since the negative is not inside the parentheses, the exponent doesn’t apply to it. A negative is also the same as multiplying by \(−1\).
   
   \(-2^{4} = (-1)(2 \times 2 \times 2 \times 2) = (-1)(16) = -16\)
   )

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