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CCSS.Math:

in this video I want to do a bunch of examples involving exponent properties but before I even do that let's have a little bit of a review of what an exponent even is so let's say I had two to the third power you might be tempted to say oh is that six and I would say no it is not six this means two times itself three times so this is going to be equal to two times two times two which is equal to 2 times 2 is 4 4 times 2 is equal to 8 if I were to ask you what 3 to the second power is or 3 squared this is equal to 3 times itself two times this is equal to 3 times 3 which is equal to 9 let's do one more of these I think you're getting the general sense if you've never seen these before let's say I have I don't know 5 to the seventh power that's equal to five times itself 7 times 5 times 5 times 5 times 5 times 5 times 5 times 5 that's seven right one two three four five six seven and this is going to be a really really really really large number and I'm not going to calculate it right now if you want to do it by hand feel free to do so or use a calculator but this is a really really really large number so one thing that you might appreciate very quickly is that exponents increase very rapidly five to the 17th would be even a way way more massive number but anyway that's a review of exponents let's get a little bit let's get a little bit steeped in algebra using exponents so what would what would 3x let me do this in a different color what would 3x times 3x times 3x B well one thing you need to remember about multiplication is it doesn't matter what order you do the multiplication in so this is going to be the same thing as 3 times 3 times 3 times X times X times X and just based on what we reviewed just here that part right there 3 times 3 3 times that's 3 to the 3rd power and this right here x times itself 3 times that's X to the 3rd power so this whole thing can be written as 3 to the 3rd times X to the 3rd or if you know what 3 to the 3rd is this is 9 times 3 which is 27 this is 27 X to the 3rd power now you might have said hey wasn't 3x times 3x times 3x wasn't that 3x to the third power right you're multiplying 3x times itself 3 times and I would say yes it is so this right here you could interpret that as 3 X to the third power and just like that we stumbled on one of our exponent properties notice this when I have some things times something and that whole thing is to the 3rd power that equals each of those things to the 3rd power times each other so 3x to the third is the same thing as 3 to the 3rd times X to the 3rd times X to the 3rd which is 27 to the 3rd power let's do a couple more examples what if I were to ask you what if I were to ask you what 6 to the 3rd times 6 to the sixth power is and this is going to be a really huge number but I want to write it as a power of 6 let me write the 6 to the 6x you in a different color 6 to the 6 times 6 6 to the 3rd times 6 to the sixth power what is this going to be equal to well 6 to the 3rd we know that 6 times itself 3 times so it's 6 times 6 times 6 and then that's going to be x the x here is in green so I'll do it in green or maybe I'll make both of them in orange that is going to be times 6 to the sixth power well what's 6 to the sixth power that's 6 times itself 6 times so it's 6 times 6 times 6 times 6 times six let's see it's 5 then you one more times 6 so what is this whole number going to be well this whole thing we're multiplying 6 times itself how many times 1 2 3 4 5 6 7 8 9 times right 3 times here and then another 6 times here so we're multiplying 6 times itself 9 times 3 plus 6 so this is equal to 6 to the 3 plus 6 power or 6 to the 9th power and just like that we've stumbled on another exponent property when we take exponents in this case 6 to the third the number 6 is of the base we're ready to taking the base to the exponent of 3 when you have the same base and you're multiplying two exponents with the same base you can add the exponents so if I have let me do several more examples of this if I have let's do it here do it in magenta let's say I had 2 squared times 2 to the fourth times 2 to the sixth well I have the same base in all of these so I can add the exponents this is going to be equal to 2 to the 2 plus 4 plus 6 which is equal to 2 to the 12th power and hopefully that makes sense because this is going to be 2 times 2 2 times itself two times 2 times itself 4 times 2 times itself 6 times so when you multiply them all out it's going to be 2 times itself 12 times or 2 to the 12th power let's do it in a little bit with a little bit more abstract way using some variables but it's the same exact idea what is X to the squared or x squared times X to the fourth well we could use the property we just learned we have the exact same base X so it's going to be X to the 2 plus 4 power it's going to be X to the sixth power and if you don't believe me what is x squared x squared is equal to x times X and if you're going to multiply that times X to the fourth you're multiplying it by x times itself four times X times X times X times X so how many times are you now multiplying X by itself well one two three four five six times X to the sixth power let's do another one of these the more examples you see I figure the better so let's do let me do the other property just to mix and match it let's say I have a to the third a to the third to the fourth power so I'll tell you the property here and then I'll show you why it makes sense when you have something to an exponent and then you raise that to an exponent you can multiply the exponents so this is going to be a to the 3 times 4 power or a to the twelfth power and why does that make sense well this right here this right here is a to the third time's itself four times so this is equal to a to the third times a to the third times a to the third times a to the third well we have the same base so we can add the exponents so this is going to be 8 to the three times four right this is equal to a to the 3 plus 3 plus 3 plus 3 power which is the same thing as a to the 3 times 4 power or a to the 12th power so just to review the properties we've learned so far in this video besides just a review of what an exponent is if I have something let's say I have X to the a power times X to the B power this is going to be equal to X to the a plus B power we saw that right here we saw that right here x squared times X to the fourth is equal to X to the 6 2 plus 4 we also saw that if I have x times y to the a power this is the same thing as X to the a power times y to the a power we saw that early on in this video we saw that over here 3x to the third is the same thing as 3 to the third times X to the that's what this is saying right here three X to the third is the same thing as three to the third times X to the third and then the last property which we just stumbled upon is if you have X to the a and then you raise that to the beat power that's equal to X to the a times B and we saw that right there a to the third and then raise that to the fourth power is the same thing as a to the 3 times 4 or a to the 12th power so let's use the properties these properties to do a handful of what we could call more complex more complex problems so let's say we had oh I don't know let's say we have let's say I want to do something a little bit let's say we have 2x y squared times negative x squared Y squared times 3x squared y squared and we wanted to simplify this well a good place to start let's see maybe we could simplify this this you could view as negative 1 times x squared times y squared so just this part right here if we take this whole thing to the squared power this is like raising each of these to the second power so this part right here could be simplified as negative 1 squared times x squared squared times y squared and then if we were to simplify that negative 1 squared is just 1 x squared squared remember you can just multiply the exponents so that's going to be X to the fourth Y squared that's what this middle part simplifies to and let's see if we can merge it with the other parts the other parts just to remember we're 2x y squared and then 3x squared y squared well now we're just going ahead and just straight up multiplying everything and we learned in multiplication that it doesn't matter which order you multiply things in so I can just rearrange we're just going in multiplying 2 times x times y squared times X to the fourth times y squared times 3 times x squared times y squared so I can rearrange this and I will rearrange it so that it's in a way that it's easy to simplify so I can multiply 2 times 3 and then I can worry about the X terms let me do it in this color then I have times x times X to the fourth times x squared times x squared and then I have to worry about the Y terms times y squared times y squared times another y squared times another y squared times another y squared and now what are these equal to well 2 times 3 you knew how to do that that's equal to 6 and what is x times X to the fourth times x squared well one thing to remember is X is the same thing as X to the first power anything to the first power is just that number so you know 2 to the first power is just 2 3 to the first power is just 3 so what is this going to be equal to this is going to be equal to we can add we have the same base X we can add the exponents X to the 1 plus 4 plus 2 power and I will add it in the next step and then on the Y's this is times y to the 2 plus 2 plus 2 power and what does that give us that gives us 6 X to the seventh power y y to the sixth power now I'll just leave you with something that you might already know and but it's pretty interesting and that's the question of what happens when you take something to the 0th power so if I say 7 to the 0 of power what does that equal and I'll tell you right now and this might seem very counterintuitive this is equal to 1 or 1 to the 0th power is also equal to 1 anything to the zeroth power any non-zero number to the 0 power is going to be equal to 1 and just to give you a little bit of an intuition on why that is think about it this way think about it this way 3 to the first power so let me write the powers 3 to the first second third will just do it with number three with the number three so 3 to the first power is 3 I think that makes sense 3 to the second power is 9 3 to the third power is 27 and of course we're trying to figure out what should three to the zeroth power be well think about it every time you you decrement the exponent every time you take the exponent down by one you are dividing by three to go from 27 to 9 you divide by 3 to go from 9 to 3 divided by 3 so we go from this exponent of that exponent maybe we should divide by 3 again and that's why anything to the zeroth power in this case 3 to 0th power is 1 see in the next video