Multiplying monomials

let's say that we wanted to multiply 5x squared and loosen purple 3 X to the fifth what would this equal pause this video and see if you can reason through that a little bit all right now let's work through this together and really all we're going to do is use properties of multiplication and use properties of exponents to essentially rewrite this expression so we can just view this if we're just multiplying a bunch of things it doesn't matter what order we multiply them in so you could just view this as five times x squared times three times X to the fifth or we could multiply our five and three first so you could view this as five times three times three times x squared times x squared times X to the fifth times X to the fifth and now what is five times three and I think you know that that is 15 now what is x squared times X to the fifth now some of you might recognize that exponent properties would come into play here if I'm multiplying two things like this we have the same base and different exponents that this is going to be equal to X to the and we add these two exponents X to the two plus five power or X to the seventh power if what I just did seems counterintuitive to you I'll just remind you what is x squared x squared is x times X and what is X to the fifth that is x times X times X times X times X and if you multiply them all together what do you get well you got seven X's and you're multiplying them all together and that is X to the seventh and so there you have it five x squared times three X to the fifth is 15 X to the seventh power so the key is is look at these coefficients look at these numbers the 5 and the 3 multiply those and then for any variable you have if you have X here so you have a common base then you can add those exponents and what we just did is known as multiplying monomials which sounds very fancy but this is a monomial monomial and in the future we'll multiplying things like polynomials where we have multiple of these things added together but that's all it is multiplying monomials let's do one more example and let's use a different variable this time just to get some variety in there let's say we want to multiply the monomial 3t to the seventh power times another monomial negative 4t pause this video and see if you can work through that alright so I'm going to do this one a little bit faster I am going to look at the three and the negative four and I'm going to multiply those first and I'm going to get a negative twelve and then if I were to want to multiply the T to the seventh times T once again they're both the variable T is our base so that's going to be T to the seventh times T to the first power that's what T is that's going to be T to the seven plus one power or T to the eighth but there you go we are done again we've just multiplied another set of monomials