Using deductive reasoning

use deductive reasoning and the distributive property to justify X plus y squared is equal to x squared plus 2xy plus y squared provide the reasoning for each step now when they say use deductive reasoning and all of the stuff it might seem like something daunting and new but this is no different than what we've done in the past in fact we've done this very exact problem so let's just do it step by step show our logic and that's essentially deductive reasoning we're starting with a statement and we're going to deduce that this is going to be equal to something else so let's start with X plus y squared let's start with X plus y squared X plus y squared we know an exponent means to multiply something by itself that many times so we know that this is the same thing or we can deduce that is the same thing as X plus y times and I'll do this next X plus y in a different color times X plus y that's what X plus y squared is now they say to use the distributive property now just as a bit of review here and we've seen it many many many times before the distributive property just tells us that if we have a times B plus C be do as many colors as possible B plus C that this is equal to this is equal to a you distribute it is equal to a times each of these terms a times B a times B plus a times C plus a times C all the tribute of property because you're distributing the a in all of the terms that in the expression that you're multiplying a by now we can do the exact same thing here instead of an A you can imagine this is an X plus y and we could take this entire X plus y and we can distribute it onto both of the terms on this expression that it is multiplying if this was an a would be ax plus ay Y now that's an X plus y we multiply the X plus y times each of those terms and that's just by the distributive property so by the distributive property that's going to be equal to distribute this on to each of them X plus y times X X plus y times X and actually I don't even have to write the X after I could just write it there doesn't matter whether you multiply X times X plus y or x plus y times X or it doesn't matter so that's that times that and then it's going to be plus y plus y times X plus y plus y times X plus y and now we can apply the distributive property again we have X being multiplied by X plus y then we have a Y being multiplied by X plus y so let's just do that again so then we get this is equal to x times X x times X plus plus x times y plus x times y I'm going through great pains to keep the colors consistent plus plus y times X plus y times this x over here plus y plus y times that Y over there I'm doing this a lot slower and I'm not skipping any steps here now what do each of these things equal x times X that is the same thing as x squared so this is equal to x squared this right here X Y we have One X Y but then we have Y X is also the same thing as X Y doesn't matter what order you multiply it in so XY plus XY is 2xy plus 2xy and then this last term right here Y times y that's the same thing as Y squared so we're done we've used deductive reasoning we've just used logical steps to start with a statement to start with an expression really and we essentially just logically manipulated it we logic we started with you can imagine it I guess you could call this a statement I guess it's the best thing to call it and we logically manipulated to come up with another statement another fact we know that if we know that this is equal to this using logical properties and distributive property and things like that in properties of exponents