# Why x^0 equals 1

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I posted this under Linda's thread, but thought I would post here in case people weren't continuing to follow that thread:

I thought about it and realized that maybe you don't even understand why any number (other than 0) raised to the 0 power = 1.

It all has to do with how exponents operate. I will explain it via a simplified proof.

When you multiply numbers with the same base that have exponents, you can see that you add the exponents.

x^2 * x^3=x^5 (you should see that by x*x=x^2 and x*x*x=x^3, and therefore altogether:

x*x*x*x*x or x^(2+3), so that is the same thing as X^(5)

When dividing by same bases with exponents, you subtract the denominator's exponent from the numerator's:

x^2/x^3 is the same thing as x*x/x*x*x you should be able to see that you can eliminate the 2 x's in the numerator, leaving 1 x in the denominator which = 1/x.

That is the same as x^(2-3) or x^(-1)......a negative exponent indicates an inverse of the number so in this case the number, x, moves to the denominator or 1/x.

So, now to prove that any number (other than 0) to the 0 power is 1.

x^6/x^6........you should be able to see that the x's are all divisible by each other, thereby leaving you with 1. (similar to 2/2, or 6/6, or 10/10) So, by the rules of exponents that I already explained, x^6/x^6 is the same thing as saying x^(6-6) or x^0. Therefore, it = 1.

I hope that helps someone!

Edited by 8FillTheHeart
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x^6/x^6........you should be able to see that the x's are all divisible by each other, thereby leaving you with 1. (similar to 2/2, or 6/6, or 10/10) So, by the rules of exponents that I already explained, x^6/x^6 is the same thing as saying x^(6-6) or x^0. Therefore, it = 1.

... and your explanation also demonstrates why 0 to the zeroth power is undefined, rather than being 1. If x = 0, then (x^n)/(x^n) would be 0/0, and any number divided by zero is undefined.

Yay math!

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... and your explanation also demonstrates why 0 to the zeroth power is undefined, rather than being 1. If x = 0, then (x^n)/(x^n) would be 0/0, and any number divided by zero is undefined.

Yay math!

I've been watching the Teaching Company Basic Math course with my 8th grader, and we learned that 0/0 is actually indeterminate rather than undefined. The reason is that the quotient can be any number because any number * 0 = 0.

If I knew that sometime in the past, I had forgotten it until this week. :tongue_smilie:

D'oh! :blushing:

You're right.

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I posted this under Linda's thread, but thought I would post here in case people weren't continuing to follow that thread:

I thought about it and realized that maybe you don't even understand why any number (other than 0) raised to the 0 power = 1.

It all has to do with how exponents operate. I will explain it via a simplified proof.

When you multiply numbers with the same base that have exponents, you can see that you add the exponents.

x^2 * x^3=x^5 (you should see that by x*x=x^2 and x*x*x=x^3, and therefore altogether:

x*x*x*x*x or x^(2+3), so that is the same thing as X^(5)

When dividing by same bases with exponents, you subtract the denominator's exponent from the numerator's:

x^2/x^3 is the same thing as x*x/x*x*x you should be able to see that you can eliminate the 2 x's in the numerator, leaving 1 x in the denominator which = 1/x.

That is the same as x^(2-3) or x^(-1)......a negative exponent indicates an inverse of the number so in this case the number, x, moves to the denominator or 1/x.

So, now to prove that any number (other than 0) to the 0 power is 1.

x^6/x^6........you should be able to see that the x's are all divisible by each other, thereby leaving you with 1. (similar to 2/2, or 6/6, or 10/10) So, by the rules of exponents that I already explained, x^6/x^6 is the same thing as saying x^(6-6) or x^0. Therefore, it = 1.

I hope that helps someone!

I just want to say how happy I am you are here. What a blessing you are...as a whole...to this board.

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:iagree: with johnandtinagilbert!

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