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# Exponent properties with products

Learn how to simplify exponents when the numbers are multiplied with each other. We'll learn that (a*b)^c is the same as a^c*b^c, a^c*a^d is same as a^(c+d) and (a^c)^d is equal to a^(c*d). We will also solve examples based on these three properties. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• Anything to zero power is one? Does that include zero to the zero power? - thanks
• at where did the -1 go ?
• (-1)^2 = +1
The set then becomes:
(+1 * x^4 * y^2)
Because you're multiplying by positive 1, the answer will not change. For this reason, the 1 is dropped entirely.
• Why is n^0 = 1? Since n^1 = n(n times itself once) and n^2 = n * n(n times itself twice), shouldn't n^0 = 0(n times itself 0, or no times)? I understand Sal's point of view on this, but I think that my explanation is right. Why doesn't mine work?
• You don't subtract n each time, you divide by n. Using that pattern:

n^2 = n * n *1, (divide by n) n^1 = n * 1, (divide by n) and n^0 = 1.

I had the same problem too for a while. If it still doesn't make sense, just do what I did for a long time before I understood:

Remember that mathematicians are slightly crazy. I only understand because I became near-mathematician-level crazy. This is not an insult. It means that you are one step closer to becoming as smart as Einstein (because you are harder to understand and only a few people could understand Einstein).
• 0 to the 0 power is undefined
anything else to the 0 power is 1.
• Thus 0 to the power 0 is undefined!
But any positive number to the power 0 is 1, so 0 to the power 0 should be 1. We can't have it both ways. Underlying this argument is the same idea as was used in the attempt to define 0 divided by 0.
• I see in the comments that this has been covered. However, don't think the explanation is satisfactory as the answer does not fit within the rules set for ALL other exponents.
So if x to the 3rd is x*x*x, x to the 2nd is x*x, and x to the first is just x. Then why would x to the zero not be 0? What is the justification to make zero mean something other than a null or nothing?
Example, take any of the above I mentioned. I ask you to write the expression x to the 3rd. Then you would write down x three times, same with the second and first.
So if you are writing down x however many times the exponent shows and if it shows zero times, then you write down nothing.
I have found nothing that justifies the breaking of the rules used for all other exponents just for a null value.
If math is logical, then what makes a zero exponent separate rule logical. This is what I could not find. Sorry for the long version for the question, but had to share my thinking.
• First, don't apologize for the long question. It's well thought out, cohesive, and my pleasure to answer.
We can determine the zero property of exponentiation by looking at ratios of exponents. If we had 2^5 / 2^4, we could write this as:
(2 * 2 * 2 * 2 * 2) / (2 * 2 * 2 * 2)
Using what we know about division, four 2's cancel out from the numerator and denominator, leaving us with just two. 2 is the same as 2^1, and the difference in the exponents that we started with was 1. Now, if we make the exponents the same, every 2 would cancel out, leaving us with 1, as x / x = 1.

We could look at taking an x away as dividing by x. If we write x three times (x*x*x), and then want to write x squared, it would be x*x*x/x, which simplifies to x*x. If we want to go from x^1 to x^0, we divide by x, just like before. x/x = 1, regardless of what x is.

Does this help with your question? It's not really a completely arbitrary rule made so that the rest of math would make sense, it works out that way. With the higher and higher abstractions of exponents and further, it becomes difficult to intuitively justify some of the rules surrounding them, especially with 0, as they aren't as often seen directly in nature.
• Is anyone else here during quarantine school? Also, I know lots of people asked about 0^0 but I didn't quite get it. Can someone explain? -thanks
• Anything raised to the power of 0 is 1, that includes 0.
• what ever happend to the (-1)^2 at ? Wouldn't it just be 1? and if so were did it go?
• (-1)^2 is 1 but in this equation, multiplying by 1 will have no effect on the outcome so it can just be dropped.
• What if the exponent has a exponent? this question keeps making me bluescreen
• I'm assuming that you mean (2^2)^2 but to do this you basically multiply the powers together and then raise the base to that number. For example, (2^2)^2, we would multiply 2 to 2, and then we would raise 2 to the 4th power. Another way to think about it is that (2^2)^2 is equal to (2^2)(2^2) which is equal to 2^4.

I hope this helped
Comment back if you have any questions:)