 All right, so let's take a look at the problem of finding a prime factorization of a number. So for example, let's say I want to find the prime factorization of 25 to the second times 6 to the third. And again, our standard disclaimer, don't watch this video. And again, the idea is that one of the main goals of our course is to develop your ability to problem-solve, and you only ever get one opportunity to solve a problem. The instant someone gives you a solution, you will never again have the opportunity to solve the problem. And this doesn't just mean the problem that you are facing, but in fact every possible variation of that problem as well. So for example, here we have the problem of finding the prime factorization of this number 25 to the second, 6 to the third. But as soon as you solve that problem, as soon as you see me solve that problem, then finding the prime factors of this or anything else is not a problem. It's a task to be solved, and you won't have a problem-solving opportunity. And again, these problem-solving opportunities are very rare, and so you have to be ready to lose an educational opportunity forever. You will never get another chance to solve this type of problem. So let's proceed. So again, we want to find the prime factorization, and maybe we're lucky, and we actually have the prime factorization of the number. Well, no, doesn't seem like we're that lucky. 25 is not prime. It's 5 times 5. 6 is also not prime. It's 2 times 3. So while this is a product, it is not a product of primes, and so it's not the prime factorization. So I want to write it as a product of prime factors, and because we know our definitions, we are able to do the mathematics. When I write 25 to the second by definition, what that means is I have two factors of 25 multiplied together. Likewise, when I write 6 to the third, that means I have three factors of 6 all multiplied together. Now, these numbers are small enough, two factors, three factors, that I can just write out what that is. And in a little bit in more complicated problems, you may want to try to find a shortcut by exercising your problem-solving abilities. But in this case, just to get to start it, well, that's 25 times 25. There's our 25 to the second. That's 6 times 6 times 6. There's our 6 to the third. And I know that each of these non-prime factors can be written as a product of prime numbers. So I'll do that. We know that 25 is 5 times 5, and 5 is prime. And because that's an equals, then anytime I see the one side, I can replace it with the other. So I see a 25 here. I'll replace it. I see another 25. I'll replace it. And so each of the 25s could be replaced with a product of primes. And the other thing that I know, 6 is 2 times 3, and 2 and 3 are both prime. So again, anytime I see a 6, I can replace it with a 2 times 3. And so I'll do all those replacements. And now I have my product of prime numbers. Now, the original expression is given in exponential notation. So it's a good practice to express our final number also in exponential notation. The answer should be in the same format as the question. So let's go ahead and do that. Arithmetic is bookkeeping. And when I write numbers in exponential notation, I tell you how many of which factors. So let's take a look at this. I have 5. Well, I have 1, 2, 3. I have 4 5s. So that's going to be expressed as 5 to the 4th. I have a 2. I have 1, 2. I have 3, 2. So that's going to be 2 to the 3rd. And I have 1, 2, 3. I have 3, 3. So that's going to be 3 to the 3rd. And so my exponential form of the product is going to be 5 to the 4th, 2 to the 3rd, and 3 to the 3rd here, here, and here. And so there's my prime factorization of my original number.