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CCSS.Math:

and I want to go over some of the other core exponent properties but they really just fall out of what we already know about exponents let's say I have two numbers a and B and I'm going to raise it to I could do it in the abstract I could raise it to the C power to the C power but I'll do it a little bit more concrete let's raise it to the fourth power what is that going to be equal to well that's going to be equal to that's going to be equal to I could write it like this copy and paste this copy and paste that's going to be equal to a B times a B times a B times a B times a B but what is that equal to well when you just multiply a bunch of numbers like this it doesn't matter what order you're going to multiply it in so you could this right over here is going to be equivalent to a times a times a times a times we have four B's as well that we're multiplying together times B times B times B times B and what is that equal to well this right over here is a to the fourth power a to the fourth power and this right over here is B B to the fourth power B to the fourth power and so you see if you take the product of two numbers and you raise them to some exponent that's equivalent to taking each of the numbers to that exponent and then taking their product and here I just used the example with four but you could do this really with any arbitrary you could do this with actually any actually any integer or actually any of any exponent you can actually this property holds and you could satisfy yourself by trying different values and using the same logic right over here but this is a general property that let me write it this way that if I have a to the B a to the B to the C power that this is going to be equal to a to the C a to the C times B to the C times B to the C times B to the C power and we'll use this throughout actually mathematics when we try to simplify things and rewrite an expression in a different way now let me introduce you another core idea here and this is the idea of raising something to some power and I'll just use the example of three and then raising that to some power what could this be simplified as well let's think about it this is the same thing as a to the third this is the same thing let me copy and paste that is a to the third times a to the third it's a to the third times a to the third and what is a to the third times so this is equal to a to the third times a to the third and that's going to be equal to a to the three plus three power we have the same base and we and so we would add and they're being multiplied they're being raised to these two exponents so it's going to be the sum of the exponents which of course is going to be equal to a a to the that's a different color a it's going to be a to the sixth power a to the sixth power so what just happened over here well I had I had to I took to a to the thirds and I multiplied them together so I took these two threes and added them together so this essentially right over here this right over here you could view this as two times three this right over here is two times two times three that's how we got the six when I raise something to one exponent and then raised it to another that's equivalent to raising the base to the product of those two exponents I just did it with this example right over here but I encourage you try other numbers to see how this works and I could do this I could do this in general I could say a to the B power and then let me copy and paste that sort of copy and then I'm going to raise that I'm going to raise that to the C power I'm going to raise that to the C power well what is that going to give me well I'm essentially going to have to take C of these so one two three I don't know how large of a number C is so I'll just do dot dot dot so dot dot dot I have C of these right over here so that's what so I have C of these so there's C of them right over there so what is that going to be equal to well that is going to be equal to a to the well for each of these C I'm going to have a B that I'm going to add together so let me write this so I'm going to have a B plus B plus B plus dot dot dot plus B and now I have C of these B's so I have C B's right over here or you could view this as a this is equal to a to the C times B power C or a you could view a to the C B power so very useful so if someone were to say what is what is 35 to the third power and then that raised to the to the seventh power well this is going obviously going to be a huge number but we can at least simplify the expression this is going to be equal to 35 to the product of these two exponents it's going to be 35 to the 3 times 7 or 35 to the 21 or to the 21st power