# Algebra 2 problem: a to the zero

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I understand a to the zero power is 1. What happens when a is a negative number? For instance, a to the zero when a is -3? Is the answer +3 or -3?

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It's also 1. All integers to the power of 0 equal 1. :)

ETA: Revised: Any integer, other than zero, raised to the 0 power equals 1.

Edited by Teachin'Mine
I was wrong!
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Anything (except 0) to the power 0 is 1.

0 to the 0 is indeterminate.

ETA: Some places seem to define 0^0 as 1. I would disagree with this.

Edited by kiana
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Anything (except 0) to the power 0 is 1.

0 to the 0 is indeterminate.

ETA: Some places seem to define 0^0 as 1. I would disagree with this.

0 to the 0 power is undefined, but if it could be defined, it would be 1.

Here's the explanation:

http://www.math.hmc.edu/funfacts/ffiles/10005.3-5.shtml

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0 to the 0 power is undefined' date=' but if it could be defined, it would be 1.

Here's the explanation:

I disagree with their assessment of the situation.

The essential reason why it is undefined is that the limit of x^y as (x,y)-> (0,0) may attain any value depending on the path chosen, hence the limit doesn't exist.

There are some circumstances (which they've listed, and done a good job of explaining) where it makes sense to "consider" it as one, but teaching either that it is one or should be one in a more general case will lead to difficulties both with indeterminate forms in l'Hospital's rule and with multivariate functions, where it certainly doesn't always equal 1.

In general, I believe that it should be avoided to teach something young (such as saying that x/0 = infinity) which will need to be untaught later.

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Saxon agrees with you Kiana. I'll revise that to any non-zero integer raised to the power of 0 equals 1.

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I understand a to the zero power is 1. What happens when a is a negative number? For instance, a to the zero when a is -3? Is the answer +3 or -3?

The key here is figuring out what is the base.

-3^0, the base is 3 which is raised to the 0, then the answer is negated, thus -3^0 = -1

(-3)^0, the base here is -3, raised to the 0, so the answer is 1

5x^0, the base is x, so we have 5(1) = 5.

(5x)^0, the base is 5x, so we have 1.

Some calculators will handle -3^0 correctly, others incorrectly, so it's good to experiment with your model if you're using the calculator.

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a^0 (a to the zero power) is the same thing as:

a/a

if you 'subtract' the exponent (hidden 1) from the numerator and the denominator you are left with 1/1. This works as long as a is a non-zero integer (so positive and negative non zero are ok).

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a^0 (a to the zero power) is the same thing as:

a/a

if you 'subtract' the exponent (hidden 1) from the numerator and the denominator you are left with 1/1. This works as long as a is a non-zero integer (so positive and negative non zero are ok).

Why does a have to be an integer? Doesn't this also work if a is any real number?

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Why does a have to be an integer? Doesn't this also work if a is any real number?

I was thinking the same thing.

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I was thinking too hard (definition of fraction)...

YES-- this works if 'a' is any REAL number as long as 'a' is not zero!

Thanks for the correction.

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From Kiana:

Most people can't solve this in 30 seconds. Can you? Say the opposite of these words: 1. Always 2. Coming 3. From 4. Take 5. Me 6. Down

Is this a trick question?

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From Kiana:

Is this a trick question?

Yes.

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Sorry again. I was the one who mentioned integers because the OP was wondering about -3 vs +3, so I mentioned that all integers raised to 0 equal 1.

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Yes.

Are these the correct answers? (I don't get the trick!)

Say the opposite of these words: 1. Always 2. Coming 3. From 4. Take 5. Me 6. Down

1. never

2. going

3. to

4. give

5. you

6. up

I feel really stupid. :confused:

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