 We've been talking about transferring information from the real number set to the exponents. What I want to really take care of, which is a huge part of just crunching number mathematics, is basically the exponents where you have rational numbers and the exponents. Remember if we took the real number set, the real number set is broken down into four subcategories and rational numbers, right? You have the natural numbers, whole numbers, integers and rational numbers. And when you transferred that over to the exponents, you had the natural numbers which were 1, 2, 3 all the way up to infinity, which cloned themselves basically. You multiply the close together, whole numbers introduce zero and anything to power zero is one. Integers introduce negative numbers in the exponents and any negative number in the exponent just flips your numbers, okay, if it's in the top goes, and the bottom goes to the top. And the rational numbers, when you have a fraction in the exponent, it takes the denominator and it makes it a rational number, right? Where it takes the root of it. So what I want to do, what I've been trying to do is come up with a bunch of examples where you deal with the rational numbers. Rational numbers and the exponent basically are radicals, basically dealing with radicals, crunching radicals. So they try to come up with a whole bunch of different examples of dealing with radicals and crunching them. And they've been really disjointed. I've looked at the videos and stuff. So I figured the best way to take care of this is probably just do one gigantic problem. And when you do one large problem, it takes care of a whole bunch of little problems in the process. It's just basically like real life. If you take one big, take care of one huge problem, a lot of little things get taken care of in that process. That's why a lot of teachers, well some teachers anyway, if you're taking a course, you'll find out that, well the more courses you take, you find out some teachers give, their tests are totally geared, it's just totally different worlds when you take a course with this person and a course with this person. This person might give you 60 little questions and another teacher might give you 10 really large problems. And hopefully those problems are not multiple choice because you get marks along the way even if you don't get the final answer correct, right? So what we're going to do is create a really large problem and then crunch that large problem to an answer. Now I have to come up with that problem, right? So what we're going to do is we're going to start off with the answer, a fairly simple answer, and then we're going to expand it. So we're going to basically reverse engineer a large problem. If you remember if you're a kid, if you're doing mazes, whenever you're doing mazes, you know you're sitting here, start here and here. You know you said there you go through your thing. If you ever get stuck, if you don't know how to do a maze, I remember when I was a kid, it's a lot easier starting from the end and going to the front, right? So that's what we're going to end up doing now. So we're going to start off at the end where, you know, the solution is for a really large problem and then reverse engineer and create a really large problem. Now what kind of problem? I just basically want our answer to be, you know, two terms. So let's say our answer is going to be two square root of five minus three A squared cubed of A squared. Now I'm making that one a little bit harder because, you know, you're going to get hard problems and that one's going to be fairly straightforward. Now one thing you've got to keep in mind is this answer can appear in a whole bunch of different ways, right? There's a, just like if you have a number four, for example, let's say, actually let's use green. If you have a number four, let's practice. So if you have a number four, is that coming out green? Green comes up. So if you have a number four, you can write the number four any which way you want. Four could be, you know, A, A divided by two, right? Four can also be five minus one, right? So all of these numbers are on top. The final answer can appear in a lot of different forms. So if, for example, if you're getting a multiple choice, if you go into an exam, write a multiple choice exam, you know, there'll be four or five answers, you know, A, B, C or D. When you solve a problem, you're not necessarily going to get specifically exactly those, the way the solutions are laid out. Sometimes you have to crunch the numbers and realize that, you know, number four is also eight over two. Now you never, you never write number four as eight over two because you always want to reduce to the lowest possible form. But what about getting more than radicals? I've said before radicals are really exponents. So your answers can appear as an exponent if you have a multiple choice exam. It can appear as an exponent or it can appear as radicals. For example, two square root five. So if you have two square root five, they will write this answer. Remember, radicals, when you don't have a number up here, that basically means the square root. So it's a cloning, it's a single plug, right? So when you go this way, it basically means whatever number is up here, it closes up once or twice. So when you go over here, this square root of two square root five can be, when two goes in here, it goes two times two, which becomes four, four times five is 20. So you can have this as a square root of 20. Now the final answer, they might write it down as the square root of 20, but they might also write it down as 20 to the power of one over two, okay? So keep this in mind where this number here can appear in multiple forms. Another way, for example, let's take this one. This is a cube root, right? So if you take these numbers inside the root symbol, it means everything that's going inside is going to clone itself three times and they're going to multiply each other, right? So when the three goes in, three times three times three is 27. When the A square goes in, you've got three A squares multiplying each other, and the rule for when you're multiplying. So right now, let's take all the stuff in, right? You've got the cube root, you've got the cube root of three. So it's calling itself three times three times three is 27. So you've got the cube root of 27. Then you have A squared multiplying itself three times. So you're going to have A squared times A squared times A squared. And you already have an A squared inside the radical. So you're going to have an A squared up here, right? A squared times A squared times A squared times A squared. All you do with multiplication is you've got exponents. You just add the exponents, right? You've got two plus two is four, plus two is six, plus two is eight. So the final answer for this is, you know what? The final version of this could be cube root of 27 A to the power of eight, right? So for this side of it, this term, they could write it like that. And again, that's not reduced form. But they could also write it in an exponent form. So let's bring this guy down. Remember, the cube root basically means one-third of this whole thing in the exponent, right? So it could be, I should not screw those dots. I should not draw all of them here so we're not getting confused with these guys. So this guy becomes 27 A to the power of eight. To the power of one-third. So your final answer might appear as 20 to the power of a half minus 27 A to the power of eight all to the power of one-third. And again, that could be reduced written in a different way, right? So it really depends on what they're testing you on. It depends on how well you're, how good you are at crunching numbers. The other way, if you can write this thing as an exponent, one other way you can write it as, do you have any room up here? We've got sort of seven room here. Another way that we can write this as would be two times, two times five to the power of a half minus eight squared A to the power of two over. So when you're, obviously this would be one line, right? That's a small location here. So whenever you're crunching numbers, what you really want to do is reduce it, take it to the lowest form possible and hopefully whatever exams you're writing, they're going to stick with one convention basically. If they're going to deal with radicals, they're going to keep the answer in the radicals, but if they want to throw little chicks here and there, what they might do is kick it into the exponents, okay? So for the radicals, what we're going to do is create a super problem. I guess we're going to create a big problem and this is what we're going to look for for the answer. Two square root of five minus three A squared cubed root of A squared, okay? Let's go find another wall and see where we can take this and how far we're going to break it down, okay? And one thing to keep in mind is every step could be its own little question to give you and we'll take care of that. We'll reduce it down, okay?