zero exponent with negative base

[lmsseattle]

Hi - just got this question on a quiz and afterward it was bugging me. I checked some websites and I'm not getting a definitive answer.

Simplify: (-9)^0

I gave the answer of 1, and when I put this in my calculator, I get 1. But when I put in -9^0, the answer is -1. Why do those ( ) make a difference? And isn't any number - regardless of the ( ) - when raised to the zero power supposed to be 1? Not -1, but 1.

Would really like some clarity on what is supposed to be an absolute cardinal rule: anything raised to the zero power is 1.

Thanks.

Lisa

 

[lmsseattle]

-9^0 is really -1 * 9^0

thanks, that helps. But why does (-9)^0 = 1? What makes the ( ) so important?

 

[Deleted member 4993]

thanks, that helps. But why does (-9)^0 = 1? What makes the ( ) so important?

\(\displaystyle (-9)^{0} \ = \ \dfrac{-9}{-9} \ = \ 1\)

\(\displaystyle -9^{0} \ = \ -\dfrac{9}{9} \ = \ -1\)

 

[lmsseattle]

\(\displaystyle (-9)^{0} \ = \ \dfrac{-9}{-9} \ = \ 1\)

\(\displaystyle -9^{0} \ = \ -\dfrac{9}{9} \ = \ -1\)

thank you. Very clear now.

 

[lmsseattle]

It is a CONVENTION of mathematical notation. Consider this example. 3² - 3² = 0, right? So - 3² is the negative, the additive inverse, of 3².
But what if I want to square - 3. If I write the square of - 3 as -3², then 3² - 3² = 9 + 9 = 18. And 18 is not zero. So to avoid ambiguity, there has been agreement among mathematicians to show the square of negative 3 as (-3)² = 9 and the negative of the square of 3 as -3² = -9. It's like why the sound tay-bell means a table rather than a car. It is an agreement on how to communicate.

thank you - that helps.