 In a previous video we reversed engineered a large exponent radical problem, right? We started off with the answer and kept on expanding and kept on expanding and until we got a fairly large problem really. And we sort of talked about it and we said we're going to go back and solve that original problem. So right now what I'm going to do is we're going to write down the big problem and go through solving it, okay? So let's just get the question up on the board here. This is the problem we had from the previous video where we started off with the solution and we reversed engineered the problem which was this, right? We wanted to come up with something that looked fairly complicated and that we knew the answer of and reverse engineering something usually you end up learning a lot more than solving the problem to begin with, right? Because you're going through the mindset of trying to make something more difficult. This sort of, as we mentioned before in a previous video, you know, when you're doing a maze, if you start from the start and you can't get to the end, start at the end, it's a lot easier getting to the start. So that's what we did. So what we're going to do now is go ahead and solve this problem. Now keep in mind, this is, you know, a grade 9, grade 10 level math here with exponents and radicals. You're not using anything special. It's just they've taken a whole bunch of different rules and put them all together or we've taken a whole bunch of rules and put them all together and created something large so you can, you can just basically solve it so you feel fairly comfortable with it. If you can solve this, it means that you pretty much got, you know, the basic rules of exponents and radicals down, okay? So let's start breaking this down and see, you know, where we can take and how we can combine like terms. Let's start off with this guy. The negatives, the negative power here flips the equation. So this goes up, okay? So this becomes 5 to the power of 3 to the power of negative 3 to the power of 1 over 6. When you have exponent to an exponent to an exponent, these guys multiply each other so you got 3 times negative 3 times 1 over 6, 3 reduces 6 down to 2. So this becomes, oh, we already dealt with the negative, sorry, there is no negative here, right? So this becomes 3 over 2. So this becomes 5, let's write it down here. This becomes 5 to the power of 3 over 2, okay? Minus, let's deal with these guys, 1,000 breaks down into 10 times 10 times 10, square root means grab two things, bring them up. So this becomes you, 16 square root 10 minus 10 square root 10, so that's our top, right? 8 is 2 times 2 times 2, 2 2s come out, multiply the 2 in the front, so this becomes 4 square root 2 minus 2 square root 2 and we can now combine it like terms. So 16 square root 10 minus 10 square root 10 is 6, oops, 6 square root 10 over 4 square root 2 minus 2 square root 2, they're like terms again, so 2 square root 2 minus. Let's deal with this guy, one way you can deal with this is you can just simplify this fraction into a, or make it into a mixed fraction, right? 3 goes into 8, 3 goes into 8 twice, what's left over is 2 over 3. This becomes 8 to the power of 2 and 2 over 3, which if you use your multiplication principle with two radicals, with two exponents if they have the same base, you can split this up, this becomes a squared times a to the power of 2 over 3, which is a squared and the 3 goes in the radical, it becomes cube root of a squared. So we just went directly from here to our simplest form of this. So this would be a squared cube root of a squared. Over here these guys are like terms, these guys can just simplify straight up, right? So this would be 9 cube root of a to the power of 5 minus 5 cube root of a to the power of 5, it just becomes 4, so this is just 4 cube root of a to the power of 5. Over here, 6 divided by 2 is 3, so this becomes a cubed, right, that becomes 3. The root of a cubed is just a, a to the negative 2 times a is going to be a to the negative 1 and a negative in the denominator, so this is just now, this is all gone and that's just a to the negative 1, oops, negative 1 and a negative power just kicks up, right? So this becomes, actually let's add that in the next step, okay? So this becomes minus 4 cube root of a to the power of 5 over 2 to the a to the power of negative 1, yep, okay? Over here, this guy is just, this guy is just radical goes there, right? And it's 5 cubed, so this becomes the square root of 5 times 5 times 5. Square root means 2 5s can come out, right? So this becomes 5 square root of 5 minus 2 reduces the 6 down to 3, 2 takes the 10 down to 5, so 3 square root 5, okay? Minus, this we can't do anything with until we simplify this, until we can combine these guys. This becomes a squared cube root of a squared minus 2 reduces the 4 down to 2, a to the power of negative 1 goes up, right? So this is 2 a right now, a to the 5 means a, a, a, a, a, you got 5 a's, you're looking for triplets, triplets come out as an a, so you got an a out here, so these guys come out, you got the cube root of a squared, you got 2 a's left inside the radical. So a times a is going to be a squared, so this is 2 a squared cube root of a squared, right? Now we can combine our like terms. This guy can add to this guy because they're both square root 5's. This guy, and I usually use different symbols, different whatever to combine different like terms, right? I underline these guys, I circle these guys, I triangle these guys, I box these guys. If you've got a lot of different like terms, you use different symbols, so kick this guy down a little bit so we can write our solution down here. So 5 square root 5 minus 3 square root 5 is going to be 2 square root of 5 and negative a squared cube root of a squared minus 2 a squared cube root of a squared, again these guys can add, so that's negative 1. If there isn't a number up front, it means 1. Negative 1 minus 2 is going to be negative 3. Negative 3 a squared cube root of a squared, okay? And that is our final solution. That is this guy simplified, okay? And again, keep this in mind. All the purples, they're all equivalent and all the red, everything we've done in the red is part, you know, a small part of the large question, of the large problem they gave you. So all these guys here are equivalent, even though they look different, okay? Everybody here is equal, everybody here is equal and all these little guys are part of the whole, you know? You can't have this, you can't go to the step without this guy. So all these little guys are our calculations for us to get to the final answer and all these things are equal and that's the beauty of mathematics where, you know, something may look completely different than something else, but once you start crunching numbers and combining like terms or bringing other equations in there, okay? Sometimes this is, we're dealing with numbers here and, you know, one variable. Sometimes you have each variable, you know, represent another function so you can bring other functions in and substitute them in and all of a sudden, you know, crazy equations reduced down to something extremely simple, you know, that, you know, some of the equations that you've heard about, you know, F equals MA, force is equal to mass times acceleration or E equals MC squared, so it's just combining, taking, you know, crazy terms and just combining things, reducing them and coming out with something that's fairly elegant, okay? This mathematics is taking, you know, trying to simplify life for us down to its basic elements, basic, you know, reduce it to a simplest term so we can deal with it more easily. And if you took each one of these, the purples, anyway, each one of the purples and punched them in your calculator, okay, the very, the answer, the numerical answer that you got, if it was a value, would be equivalent, they're all the same.