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### Course: 7th grade>Unit 5

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?

• 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!
• At how does 1^0 equal 1?
• Anything to the 0 power is equal to 1 unless you do 0^0.
• Why is 2^0 equal to one?
• Any non-zero number to the power of 0 is always equal to one. Here's a way I like to think about it:

You have 8x^2 apples, 7x^1 apples, 6x^1 apples and 5 apples. Now, since the 5 apples have no 'x' variable inside, you can express that term as 5x^0. If x^0 was 0, then 5 x 0 would be 0, and the 5 apples would just disappear. For this to mathematically work, you could only make x^0 (x=non-zero number) equal to 1!
• Wouldn’t it be useful to also simply define any number raised to the zero power as 1? The given justifications rely on two definitions any way. Why not use one definition instead of two?
• i don't understand how anything to the first power would be 1. it seems like it would be, like if 4 to the first power would mean that you have one set of four meaning the answer would be four.
• You are correct. 4^1 = 4, but that's not what this video is about. This video is saying 1^x = 1 for all x, and that 0^x = 0 for all non-negative x
• What do you do with a negative exponent how do you solve a question like [(4 to the negative power of 3 )times 5]
• When you take numbers to a negative exponent, you use this idea: If b is any number, then b to the negative n power (where n is also any number) is 1/b to the absolute value of n. If this is confusing, take your example: If you have 4 to the negative -3 power, then the answer would become 1/(4 to the 3 power), which would be 1/4*4*4, or 1/64.
• I still don't get why 2^0 = 1
Is there any reason why you have to start with a 1 when multiplying exponents? Why not start with 2 or 10?
• 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.
• 1,000,000 has seven digits, 999,999 has six digits.
How is it that 1 million is an even number while 999,999 is odd? Shouldn't it be inverted or am I missing something?
(1 vote)
• it doesn't matter how many digits a number has, as long as it can be evenly divided by two it's even and the ones that can't are odd :)
• What is 0^0 actually equal to?
• 0^0 is undefined. You can find the video on powers of 0 by using the search bar on any KA screen.