Alright, today I gonna show you how to prove a math identity, by Art and Craft! What’s up everyone, thank you for spending a little bit of your time with me today, and if this is your first time here, welcome, and please subscribe, so you will not miss my next video. alright, today we gonna prove two math identities, which are a^3+b^3, and also a^3-b^3. So, what’s the motivation behind? Think of it this way, if you have a cube, each side has length A, Then the volume of that cube is just a^3. If the length is b, the volume is b^3. Then you take those two cubes together, you combine them. Then the volume of the whole thing gonna be a^3+b^3. So, we gonna do that, LITERALLY! Meaning, we gonna build a cube, which each side has length a, another cube, which each side has length b, we combine them, and compute the volume at the end. and, we gonna do the same for a^3-b^3. And, the way we gonna do that is, I gonna follow my professionally drawn diagram, like this. And, let’s see what happens! alright, so I have my first cube, this gonna be cube of size b. So I have a lot more to go. Let’s keep working! alright, second cube. oh alright, third one! Alright, third one I mean, fourth one. So, if I do it right, this should kind of form one big box together, something like this. Alright, we have to do one more. But this last one is actually too big for one piece of paper, so I need to cut, maybe two pieces of paper, and combine them together, so let’s go. Alright, last box is done! So, I have cut up all the boxes already, now I gonna have to pack up and move somewhere else, because I am out of time in this room already. So, I have to pack up and, see you in a bit. Now, we are ready, so let’s do prove to those two identities. So, let’s start off with a^3-b^3. So, a gonna be the length of the big cube. b gonna be the length of the small cube. the whole volume gonna be a^3. Therefore, if I take out b^3, the remaining volume will be a^3-b^3 with this, I can rearrange the boxes in the following way. I put this on top, and put the small box to the side. Now, we can try to find the volume of this solid. The formula that we gonna be using is we gonna this of this part as the base. and this part is the height. So, the volume of this solid gonna be the area of the base times height. What’s the area of the base? This length is a, this length is b This length is a. Area of this one gonna be (a+b) times a, which is a^2+ab. And, area of this guy gonna be b^2. So, the total area of the base gonna be a^2+ab+b^2 The height gonna be simply a-b. So, we have the formula of the solid being (a-b) times a^2+ab+b^2 but remember the volume of is this is a^3-b^3. So, we conclude that a^3-b^3 is (a-b)(a^2+ab+b^2). Let’s go over another one. Again, we start of with a^3, Now, we gonna add b^3 to it Now, the total volume is a^3+b^3. I can rearrange the boxes this way. Again, this gonna be base times height. Base gonna be this guy. Height gonna be this. Height, this time, gonna be a+b. How about the base? This length right here is a. This length right here is (a-b). So, the big part here, has the area of a(a-b). which is a^2-ab. This little small part is b^2. So, the total area of the base gonna be a^2-ab+b^2. So, the total volume gonna be (a+b) times a^2 – ab + b^2 But the total volume, who know from earlier, it’s a^3+b^3. Therefore we conclude that a^3+b^3 is the same thing as (a+b)(a^2-ab+b^2) alright, for those of you who are interested, I have made templates for all the boxes So, you can just print it out, cut it, and follow instructions, And you gonna have boxes just like I have. You can find the link to the template in the description down below. Alright, that’s all I want to talk about today, So, this video has been pretty fun for me. even though, I hate craft so much, but math part is pretty fun, I think it’s pretty cool. And if you like this video too, don’t forget to leave a like down below. and if you have not done so already, subscribe so you will not miss my next video. For today, thank you for watching, my name is Kuang, and you are watching n choose k. Bye Bye!