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Lesson 7: Powers with rational bases

# 1 and -1 to different powers

Different exponents affect the value of a number: when raised to the power of zero, any number equals one; when raised to an even power, negative numbers yield positive results; and when raised to an odd power, negative numbers yield negative results. Created by Sal Khan.

## Want to join the conversation?

• At how does 1^0 equal 1?
• Anything to the 0 power is equal to 1 unless you do 0^0.
• plz try to explain it as if you were explaining this to a small kid plz that might help me
• I think Sal making the video longer is what's confusing us! But it's actually pretty simple! One to the power of ANY NUMBER will always be one because you just keep on doing 1 x 1 x 1 x 1 etc. When you do -1 to the power of an ODD number, the answer is always -1, but when you do -1 to the power of an even number, the result is always just 1! :) Let me know if this helps!
• Is zero an even or odd power?
• 0 is an even number and is hence an even power.
• Wait if he did 2^4 it also means 2x2x2x2 and so if that is it why is there to put a 1 it is just going to be the same answer
• There are two different ways to "think" of the calculation of the exponent.

The first is to multiply the number by itself as many times as the exponent says to do so. Example:

5^3 is calculated as: 5x5x5=125

The other way to picture the calculation of an exponent is to start from the number one and then multiply as the exponent says to. Example:

5^3 is calculated as: 1x5x5x5=125

This doesn't impact the process much. Only if you happen to be going to the power of zero, which is why everthing to the power of zero is 1. Although both proceses are correct which one is more correct is up for debate. As such untill we get a definitive answer best to learn both ways.
• Isn't it easier to do the math without the 1 at ?
• I think he's trying to get us in the habit of using the 1 for when you need to, like when multiplying to the 0 power.
• the pattern is: if its an odd number of digits, it will be negative. if its even the results will be positive. works only if there are paranthesis () these things
• Yes, if the base of the exponent is negative, and the base has parenthesis, the exponent can determine whether the outcome is positive or negative.

An odd exponent (ex -1, -3, -9) will have a negative answer, however, an even exponent will result in a positive answer.

Note that the outcome can only be negative if one or more of the following are true:

-The Base is Negative and the exponent is odd
-The Exponent is Negative and odd
• at is it just like some random person is going to walk up to us and ask 'wHaT iS oNE tO tHe onE mIllIonTHe pOWeR?!' i highly doubt it.
• I still don't get why 2^0 = 1
(1 vote)
• I think the simplest way to understand it is this. Start by taking some powers of 2:

2^2 = 4
2^3 = 8
2^4 = 16
...

Notice, based on this, that it is pretty simple to go forwards from one power to the next. If you want to go from "2^2" to "2^3", just multiply by 2. For instance:

2^2 * 2 = 2^3

But what if we wanted to go backwards? For instance, what if we wanted to go from 2^4 backwards to 2^3? Simple: we divide.

2^3 / 2 = 2^2

So now, what if we wanted to find 2^0? Well, simple! Start with 2^1, and go backwards. How do we go backwards again? We divide!

2^1 / 2 = 2^0

And "2^1 / 2" is just 1. So that's why "1 = 2^0".

And as for the other part of your question, the reason we start from 1 is because it makes the math work. For instance, if you said that "2^0 = 10", then "2^1" would be 10 * 2, which would be 20. And "2^3" would be "10 * 2^2", which would be 40. And if we do that, then what's the point of exponents? Exponents are supposed to be used to multiply the number by itself. But if we throw a 10 in there, then exponents are pretty useless. However, it is okay for us to multiply by 1, because multiplying by 1 does nothing; it doesn't change anything. So that's why we always start with 1.

Anyways, I hope this helped a bit.
• What if you multiply or square something by one and its a negative one would the outcome be negative
• If you raise a negative number to an even power you get a positive number.
And if you raise a negative number to an odd power, you get a negative number.

`(-1)^1 = -1(-1)^2 = 1(-1)^3 = -1(-1)^4 = 1(-1)^5 = -1(-1)^6 = 1`
Hope that helps!