WEBVTT

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Write the prime factorization
of 75.

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Write your answer using
exponential notation.

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So we have a couple of
interesting things here.

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Prime factorization, and they
say exponential notation.

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We'll worry about the
exponential notation later.

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

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prime number?

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And just as a refresher, a
prime number is a number

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

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prime numbers-- let me write
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
by 1 and 2.

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3 is another prime number.

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Now, 4 is not prime,
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
by 1 and 5, so 5 is prime.

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

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I think you get the
general idea.

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You move to 7, 7 is prime.

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It's only divisible
by 1 and 7.

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8 is not prime.

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

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also divisible by 3,
so 9 is not prime.

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Prime is not the same thing
as odd numbers.

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Then if you move to 10,
10 is also not prime,

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divisible by 2 and 5.

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

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then a prime number.

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And we could keep going
on like this.

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People have written computer
programs looking for the

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highest prime and all of that.

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So now that we know what
a prime is, a prime

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factorization is breaking up
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
75, and I'm going to do it

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using what we call a
factorization tree.

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

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will go into 75.

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Now, the smallest prime
number is 2.

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Does 2 go into 75?

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Well, 75 is an odd number, or
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
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
3 will go into it.

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So 75 is 3 times
something else.

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

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have three quarters, you have
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
if you don't believe me.

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Multiple out 3 times 25.

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

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

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by 2 either.

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But maybe 25 is divisible
by 3 again.

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

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7 is not divisible by 3, so 25
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
factorization because now we

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have all prime numbers here.

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So we can write that 75
is 3 times 5 times 5.

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So 75 is equal to 3
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
factorization, but they want

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us to write our answer using
exponential notation.

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So that just means, if we have
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
by itself two times.

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This is the same thing as
5 to the second power.

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So if we want to write our
answer using exponential

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

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