0:00:00.740,0:00:04.160
Write the prime factorization[br]of 75.

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Write your answer using[br]exponential notation.

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So we have a couple of[br]interesting things here.

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Prime factorization, and they[br]say exponential notation.

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We'll worry about the[br]exponential notation later.

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So the first thing we have to[br]worry about is what is even a

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prime number?

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And just as a refresher, a[br]prime number is a number

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that's only divisible by itself[br]and one, so examples of

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prime numbers-- let me write[br]some numbers down.

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Prime, not prime. Not prime.

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So 2 is a prime number.

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It's only divisible[br]by 1 and 2.

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3 is another prime number.

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Now, 4 is not prime,[br]because this is

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divisible by 1, 2 and 4.

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We could keep going.

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5, well, 5 is only divisible[br]by 1 and 5, so 5 is prime.

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6 is not prime, because it's[br]divisible by 2 and 3.

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I think you get the[br]general idea.

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You move to 7, 7 is prime.

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It's only divisible[br]by 1 and 7.

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8 is not prime.

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9 you might be tempted to say[br]is prime, but remember, it's

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also divisible by 3,[br]so 9 is not prime.

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Prime is not the same thing[br]as odd numbers.

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Then if you move to 10,[br]10 is also not prime,

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divisible by 2 and 5.

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11, it's only divisible[br]by 1 and 11, so 11 is

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then a prime number.

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And we could keep going[br]on like this.

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People have written computer[br]programs looking for the

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highest prime and all of that.

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So now that we know what[br]a prime is, a prime

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factorization is breaking up[br]a number, like 75, into a

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product of prime numbers.

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So let's try to do that.

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So we're going to start with[br]75, and I'm going to do it

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using what we call a[br]factorization tree.

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So we first try to find just the[br]smallest prime number that

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will go into 75.

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Now, the smallest prime[br]number is 2.

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Does 2 go into 75?

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Well, 75 is an odd number, or[br]the number in the ones place,

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this 5, is an odd number.

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5 is not divisible by 2, so[br]2 will not go into 75.

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So then we could try 3.

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Does 3 go into 75?

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Well, 7 plus 5 is 12.

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12 is divisible by 3, so[br]3 will go into it.

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So 75 is 3 times[br]something else.

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And if you've ever dealt with[br]change, you know that if you

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have three quarters, you have[br]75 cents, or if you have 3

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times 25, you have 75.

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So this is 3 times 25.

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And you can multiply this out[br]if you don't believe me.

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Multiple out 3 times 25.

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Now, is 25 divisible by--[br]you can give up on 2.

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If 75 wasn't divisible by 2,[br]25's not going to be divisible

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by 2 either.

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But maybe 25 is divisible[br]by 3 again.

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So if you take the digits[br]2 plus 5, you get 7.

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7 is not divisible by 3, so 25[br]will not be divisible by 3.

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So we keep moving up: 5.

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Is 25 divisible by 5?

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Well, sure.

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It's 5 times 5.

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So 25 is 5 times 5.

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And we're done with our prime[br]factorization because now we

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have all prime numbers here.

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So we can write that 75[br]is 3 times 5 times 5.

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So 75 is equal to 3[br]times 5 times 5.

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We can say it's 3 times 25.

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25 is 5 times 5.

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3 times 25, 25 is 5 times 5.

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So this is a prime[br]factorization, but they want

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us to write our answer using[br]exponential notation.

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So that just means, if we have[br]repeated primes, we can write

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those as an exponent.

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So what is 5 times 5?

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5 times 5 is 5 multiplied[br]by itself two times.

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This is the same thing as[br]5 to the second power.

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So if we want to write our[br]answer using exponential

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notation, we could say this is[br]equal to 3 times 5 to the

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second power, which is the[br]same thing as 5 times 5.