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# Negative exponent intuition

CCSS.Math:

## Video transcript

I have been asked for some intuition as to why let's say a to the minus B is equal to one over a to the B and before I give you the intuition I want you to just realize that this really is a definition the I don't know the inventor of mathematics you know wasn't one person it was you know a convention that arose but they did they define this and they define this for the reasons that I'm going to show you well the what I'm going to show you is one of the reasons and we'll see that this is a good definition because once you learn to exponent rules all of the other exponent rules stay consistent when you for negative exponents and when you raise something to the zeroth power so let's take the positive exponents those are pretty intuitive I think so the positive exponents so you have a to the 1 a squared a cubed a to the fourth what's a to the 1 a to the 1 we said was a and then to get to a squared what did we do we multiply it by a right a squared is just a times a and then to get to a cubed what did we do we multiplied by a again and then to get to a to the fourth what do we do we multiplied by a again or the other way you could imagine is when you decrease the exponent what are we doing we are multiplying by 1 over a or dividing by a it's similarly decrease again you're dividing by a and go from a squared to a to the first you're dividing by a so let's use this progression to figure out what a to the 0 is so this is the first hard one so a to the 0 so you're the you're the inventor the founding mother of mathematics and you need to define what a to the 0 is and you know maybe it's 17 maybe it's some you know maybe it's pi I don't know you it's up to you to decide what a 2-0 is but wouldn't it be nice if a to the 0 retained this pattern that every time you decrease the exponent you're dividing by a right so if you're going from a to the first to a to the zero would it be nice if we just divided by a so let's do that so if we go from a to the first which is just a and divided by a right so we're just going to go we're just going to divide it by a what is a divided by a well it's just one so that's where the definition or that's one of the intuitions behind why something to the zeroth power is equal to one because when you take that that number and you divide it by itself one more time you just get one so that's pretty reasonable but now let's go into the negative domain so what should a to the negative one equal a to the negative one well once again it's nice if we can retain this pattern where every time we decrease the exponent we're dividing by a so let's divide by a again so 1 over a so we're going to take a to the zero and divide it by a a to the zero divided by a a to the zero is one so what's 1 divided by a it's 1 over a and let's do it one more time and then I think you're going to get the pattern well I think you probably already got the pattern what's a to the minus 2 well we want you know it'd be silly now to to change this pattern every time we decrease the exponent we're dividing by a so to go from a to the minus 1 to a to the minus 2 let's just divide by a again and what do we get we get if you take 1 over a and divide by a you get 1 over a squared and you could just keep doing this pattern all the way to the left and you would get a to the minus B is equal to 1 over a to the B hopefully that gave you a little intuition as to why well first of all you know the big mystery is well you know something to a zeroth power why does that equal one first keep in mind that that's just a definition someone decided that it should be equal one but they had a good reason and the good reason was they wanted to keep this pattern going and that's the same reason why they defined negative exponents in this way and what's extra cool about it is not only does it retain this pattern of when you decrease exponents you're dividing by a or when you're increasing exponents you're multiplying by a but as you'll see in the exponent rules videos all of the exponent rules hold all of the exponent rules are consistent with this definition of something to the zeroth power and this definition of something to the negative our hopefully that didn't confuse you and gave you a little bit of intuition and demystified something that frankly is quite mystifying the first time you learn it