Prime factorization

Write the prime factorization of 75. Write your answer using exponential notation. So we have a couple of interesting things here. Prime factorization, and they say exponential notation. We'll worry about the exponential notation later. So the first thing we have to worry about is what is even a prime number? And just as a refresher, a prime number is a number that's only divisible by itself and one, so examples of prime numbers-- let me write some numbers down. Prime, not prime. So 2 is a prime number. It's only divisible by 1 and 2. 3 is another prime number. Now, 4 is not prime, because this is divisible by 1, 2 and 4. We could keep going. 5, well, 5 is only divisible by 1 and 5, so 5 is prime. 6 is not prime, because it's divisible by 2 and 3. I think you get the general idea. You move to 7, 7 is prime. It's only divisible by 1 and 7. 8 is not prime. 9 you might be tempted to say is prime, but remember, it's also divisible by 3, so 9 is not prime. Prime is not the same thing as odd numbers. Then if you move to 10, 10 is also not prime, divisible by 2 and 5. 11, it's only divisible by 1 and 11, so 11 is then a prime number. And we could keep going on like this. People have written computer programs looking for the highest prime and all of that. So now that we know what a prime is, a prime factorization is breaking up a number, like 75, into a product of prime numbers. So let's try to do that. So we're going to start with 75, and I'm going to do it using what we call a factorization tree. So we first try to find just the smallest prime number that will go into 75. Now, the smallest prime number is 2. Does 2 go into 75? Well, 75 is an odd number, or the number in the ones place, this 5, is an odd number. 5 is not divisible by 2, so 2 will not go into 75. So then we could try 3. Does 3 go into 75? Well, 7 plus 5 is 12. 12 is divisible by 3, so 3 will go into it. So 75 is 3 times something else. And if you've ever dealt with change, you know that if you have three quarters, you have 75 cents, or if you have 3 times 25, you have 75. So this is 3 times 25. And you can multiply this out if you don't believe me. Multiple out 3 times 25. Now, is 25 divisible by-- you can give up on 2. If 75 wasn't divisible by 2, 25's not going to be divisible by 2 either. But maybe 25 is divisible by 3 again. So if you take the digits 2 plus 5, you get 7. 7 is not divisible by 3, so 25 will not be divisible by 3. So we keep moving up: 5. Is 25 divisible by 5? Well, sure. It's 5 times 5. So 25 is 5 times 5. And we're done with our prime factorization because now we have all prime numbers here. So we can write that 75 is 3 times 5 times 5. So 75 is equal to 3 times 5 times 5. We can say it's 3 times 25. 25 is 5 times 5. 3 times 25, 25 is 5 times 5. So this is a prime factorization, but they want us to write our answer using exponential notation. So that just means, if we have repeated primes, we can write those as an exponent. So what is 5 times 5? 5 times 5 is 5 multiplied by itself two times. This is the same thing as 5 to the second power. So if we want to write our answer using exponential notation, we could say this is equal to 3 times 5 to the second power, which is the same thing as 5 times 5.