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### Course: Algebra basics>Unit 6

Lesson 1: Exponent properties intro

# Exponent properties with products

To simplify expressions with exponents, there are a few properties that may help. One is that when two numbers with the same base are multiplied, the exponents can be added. Another is that when a number with an exponent is raised to another exponent, the exponents can be multiplied. 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).
• 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.
• 0 to the 0 power is undefined
anything else to the 0 power is 1.
• In this calculation, n will = the number that is being "powered"
Examples
n^1 = 1*n
n^2 = 1*n*n
n^3 = 1*n*n*n
n^... = 1*n*n*n*n...
and so on...

Therefore if you were to do if n^0, it would = 1, as 1 is always at the beginning.

Although, the proof of why 1 is there is not defined, many people think it is a helpful way of understanding exponents.
n^0 = 1

Because of this, 0^0 = 1, as the above equations show us.

In other words, what is 0^0? Answer: Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values.
Since n^0 is 1 for all numbers n other than 0, it would be logical to define that 0^0 = 1.
Although, these may be the only proven way of telling us 0^0 = 1, 0^0 is usually undefined, there are many debates about the topic of what 0^0 equals.

`Link: https://www.homeschoolmath.net/teaching/zero-exponent-proof.php#:~:text=In%20other%20words%2C%20what%20is,define%20that%2000%20%3D%201.`

*Hope this helps!*
(1 vote)
• 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.
• Why do the videos have to be so long
• In the last video, Sal said 0^0 is undefined. The Khan Academy calculator says it is 1. Which is correct? IM confused.
• The debate still goes on as to whether 0^0 is undefined/indeterminate or simply 1. Many different fields of math define 0^0 as 1 but others say that it is indeterminate.