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Welcome to the presentation on level one exponent rules.

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Let's get started with some problems.

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So if I were to ask you what 2 -- that's a little fatter than

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I wanted it to be, but let's keep it fat so it doesn't look

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strange -- 2 the third times -- and dot is another way of

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saying times -- if I were to ask you what 2 to the third

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times 2 to the fifth is, how would you figure that out?

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Actually, let me use a skinnier pen because that does look bad.

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So, 2 to the third times 2 to the fifth.

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Well there's one way that I think you do know how to do it.

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You could figure out that 2 to the third is 9, and

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that 2 to the fifth is 32.

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And then you could multiply them.

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And 8 times 32 is 240, plus it's 256, right?

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You could do it that way.

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That's reasonable because it's not that hard to figure out 2

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to the third is and what 2 to the fifth is.

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But if those were much larger numbers this method might

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become a little difficult.

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So I'm going to show you using exponent rules you can actually

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multiply exponentials or exponent numbers without

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actually having to do as much arithmetic or actually you

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could handle numbers much larger than your normal math

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skills might allow you to.

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So let's just think what 2 to the third times

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2 to the fifth means.

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2 to the third is 2 times 2 times 2, right?

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And we're multiplying that times 2 to the fifth.

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And that's 2 times 2 times 2 times 2 times 2.

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So what do we have here?

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We have 2 times 2 times 2, times 2 times 2 times

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2 times 2 times 2.

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Really all we're doing is we're multiplying 2 how many times?

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Well, one, two, three, four, five, six, seven, eight.

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So that's the same thing as 2 to the eighth.

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Interesting.

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3 plus 5 is equal to 8.

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And that makes sense because 2 to the 3 is 2 multiplying by

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itself three times, to the fifth is 2 multiplying by

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itself five times, and then we're multiplying the two, so

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we're going to multiply 2 eight times.

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I hope I achieved my goal of confusing you just now.

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Let's do another one.

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If I said 7 squared times 7 to the fourth.

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That's a 4.

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Well, that equals 7 times 7, right, that's 7 squared,

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times and now let's do 7 to the fourth.

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7 times 7 times 7 times 7.

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Well now we're multiplying 7 by itself six times, so

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that equal 7 to the sixth.

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So in general, whenever I'm multiplying exponents of the

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same base, that's key, I can just add the exponents.

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So 7 to the hundredth power times 7 to the fiftieth

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power, and notice this is an example now.

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It would be very hard without a computer to figure out what

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7 to the hundredth power is.

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And likewise, very hard without a computer to figure out what

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7 to the fiftieth power is.

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But we could say that this is equal to 7 to the 100 plus 50,

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which is equal to 7 to the 150.

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Now I just want to give you a little bit of warning, make

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sure that you're multiplying.

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Because if I had 7 to the 100 plus 7 to the 50, there's

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actually very little I could do here.

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I couldn't simplify this number.

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But I'll throw out one to you.

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If I had 2 to the 8 times 2 to the 20, we know we

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can add these exponents.

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So that gives you 2 to the 28, right?

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What if I had 2 to the 8 plus 2 to the 8?

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This is a bit of a trick question.

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Well I just said if we're adding, we can't

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really do anything.

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We can't really simplify it.

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But there's a little trick here that we actually have

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two 2 to the 8, right?

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There's 2 to the 8 times 1, 2 to the 8 times 2.

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So this is the same thing as 2 times 2 to the 8, isn't it?

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2 times 2 to the 8.

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That's just 2 to the 8 plus itself.

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And 2 times to the 8, well that's the same thing as 2 to

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the first times 2 to the 8.

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And 2 to the first times 2 to the 8 by the same rule we just

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did is equal to 2 to the 9.

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So I thought I would just throw that out to you.

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And it works even with negative exponents.

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If I were to say 5 to the negative 100 times 3 to the,

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say, 100 -- oh sorry, times 5 -- this has to be a 5.

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I don't know what my brain was doing.

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5 to the negative 100 times 5 to the 102, that would

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equal 5 squared, right?

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I just take minus 100 plus 102.

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This is a 5.

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I'm sorry for that brain malfunction.

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And of course, that equals 25.

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So that's the first exponent rule.

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Now I'm going to show you another one, and it kind of

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leads from the same thing.

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If I were to ask you what 2 to the 9 over 2 to the 10 equals,

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that looks like that could be a little confusing.

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But it actually turns out to be the same rule, because what's

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another way of writing this?

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Well, we know that this is also the same thing as 2 to the 9

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times 1 over 2 to the 10, right?

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And we know 1 over 2 to the 10.

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Well, you could re-write right this as 2 the 9 times 2 to

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the negative 10, right?

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All I did is I took 1 over 2 to the 10 and I flipped it and I

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made the exponent negative.

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And I think you know that already from

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level two exponents.

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And now, once again, we can just add the exponents.

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9 plus negative 10 equals 2 to the negative 1, or we could

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say that equals 1/2, right?

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So it's an interesting thing here.

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Whatever is the bottom exponent, you could put it in

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the numerator like we did here, but turn it into a negative.

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So that leads us to the second exponent rule, simplification

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is we could just say that this equals 2 to the 9 minus 10,

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which equals 2 to the negative 1.

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Let's do another problem like that.

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If I said 10 to the 200 over 10 to the 50, well that

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just equals 10 to the 200 minus 50, which is 150.

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Likewise, if I had 7 to the fortieth power over 7 to

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the negative fifth power, this will equal 7 to the

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fortieth minus negative 5.

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So it equals 7 to the forty-fifth.

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Now I want you to think about that, does that make sense?

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Well, we could have re-written this equation as 7 to the

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fortieth times 7 to the fifth, right?

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We could have taken this 1 over 7 to the negative 5 and turn it

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into 7 to the fifth, and that would also just be 7

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to the forty-five.

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So the second exponent rule I just taught you actually is no

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different than that first one.

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If the exponent is in the denominator, and of course, it

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has to be the same base and you're dividing, you subtract

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it from the exponent in the numerator.

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If they're both in the numerator, as in this case, 7

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to the fortieth times 7 to the fifth -- actually there's no

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numerator, but they're essentially multiplying by each

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other, and of course, you have to have the same base.

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Then you add the exponents.

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I'm going to add one variation of this, and actually this is

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the same thing but it's a little bit of a trick question.

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What is 2 to the 9 times 4 to the 100?

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Actually, maybe I shouldn't teach this to you, you have

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to wait until I teach you the next rule.

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But I'll give you a little hint.

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This is the same thing as 2 the 9 times 2 squared to the 100.

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And the rule I'm going to teach you now is that when you have

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something to an exponent and then that number raised to

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an exponent, you actually multiply these two exponents.

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So this would be 2 the 9 times 2 to the 200.

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And by that first rule we learned, this would

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be 2 to the 209.

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Now in the next module I'm going to cover

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this in more detail.

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I think I might have just confused you.

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But watch the next video and then after the next video I

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think you're going to be ready to do level one exponent rules.

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Have fun.

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