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Welcome to Part 2 on the presentation on Level

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1 exponent rules.

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So let's start off by reviewing the rules

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we've learned already.

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If I had 2 to the tenth times 2 to the fifth, we learned that

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since we're multiplying exponents with the same base,

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we can add the exponent, so this equals 2 to the fifteenth.

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We also learned that if it was 2 to the tenth over 2 to the

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fifth, we would actually subtract the exponents.

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So this would be 2 to the 10 minus 5, which

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equals 2 to the fifth.

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At the end of the last presentation, and I probably

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shouldn't have introduced it so fast, I introduced

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a new concept.

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What happens if I have 2 to the tenth to the fifth power?

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Well, let's think about what that means.

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When I raise something to the fifth power, that's just like

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saying 2 to the tenth times 2 to the tenth times 2 to the

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tenth times 2 to the tenth times 2 to the tenth, right?

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All I did is I took 2 to the tenth and I multiplied

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it by itself five times.

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That's the fifth power.

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Well, we know from this rule up here that we can add these

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exponents because they're all the same base.

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So if we add 10 plus 10 plus 10 plus 10 plus

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10, what do we get?

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Right, we get 2 to the fiftieth power.

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So essentially, what did we do here?

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All we did is we multiplied 10 times 5 to get 50.

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So that's our third exponent rule, that when I raise an

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exponent to a power and then I raise that whole expression to

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another power, I can multiply those two exponents.

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So let me give you another example.

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If I said 3 to the 7, and all of that to the negative 9, once

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again, all I do is I multiply the 7 and the negative 9, and

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I get 3 to the minus 63.

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So, you see, it works just as easily with negative numbers.

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So now, I'm going to teach you one final exponent property.

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Let's say I have 2 times 9, and I raise that whole thing

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to the hundredth power.

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It turns out of this is equal to 2 to the hundredth power

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times 9 to the hundredth power.

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Now let's make sure that that makes sense.

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Let's do it with a smaller example.

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What if it was 4 times 5 to the third power?

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Well, that would just be equal to 4 times 5 times 4 times 5

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times 4 times 5, right, which is the same thing as 4 times 4

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times 4 times 5 times 5 times 5, right?

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I just switched the order in which I'm multiplying, which

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you can do with multiplication.

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Well, 4 times 4 times 4, well, that's just equal

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to 4 to the third.

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And 5 times 5 times 5 is equal to 5 to the third.

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Hope that gives you a good intuition of why this

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property here is true.

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And actually, when I had first learned exponent rules, I would

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always forget the rules, and I would always do this proof

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myself, or the other proofs.

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And a proof is just an explanation of why the rule

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works, just to make sure that I was doing it right.

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So given everything that we've learned to now-- actually, let

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me review all of the rules again.

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If I have 2 to the seventh times 2 to the third,

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well, then I can add the exponents, 2 to the tenth.

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If I have 2 the seventh over 2 the third, well, here I

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subtract the exponents, and I get 2 to the fourth.

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If I have 2 to the seventh to the third power, well, here

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I multiplied the exponents.

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That gives you 2 to the 21.

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And if I had 2 times 7 to the third power, well, that equals

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2 to the third times 7 to the third.

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Now, let's use all of these rules we've learned to actually

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try to do some, what I would call, composite problems that

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involve you using multiple rules at the same time.

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And a good composite problem was that problem that I had

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introduced you to at the end of that last seminar.

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Let's say I have 3 squared times 9 to the eighth power,

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and all of that I'm going to raise to the negative 2 power.

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So what can I do here?

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Well, 3 and 9 are two separate bases, but 9 can actually

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be expressed as an exponent of 3, right?

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9 is the same thing as 3 squared, so let's

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rewrite 9 like that.

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That's equivalent to 3 squared times-- 9 is the same thing as

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3 squared to the eighth power, and then all of that to the

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negative 2 power, right?

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All I did is I replaced 9 with 3 squared because

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we know 3 times 3 is 9.

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Well, now we can use the multiplication rule on

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this to simplify it.

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So this is equal to 3 squared times 3 to the 2 times 8,

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which is 16, and all of that to the negative 2.

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Now, we can use the first rule.

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We have the same base, so we can add the exponents, and

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we're multiplying them, so that equals 3 to the eighteen power,

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right, 2 plus 16, and all that to the negative 2.

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And now we're almost done.

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We can once again use this multiplication rule, and we

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could say 3-- this is equal to 3 to the eighteenth times

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negative 2, so that's 3 to the minus 36.

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So this problem might have seemed pretty daunting at

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first, but there aren't that many rules, and all you have to

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do is keep seeing, oh, wow, that little part of the

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problem, I can simplify it.

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Then you simplify it, and you'll see that you can keep

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using rules until you get to a much simpler answer.

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And actually the Level 1 problems don't even involve

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problems this difficult.

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This'll be more on the exponent rules, Level 2.

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But I think at this point you're ready

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to try the problems.

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I'm kind of divided whether I want you to memorize the rules

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because I think it's better to almost forget the rules and

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have to prove it to yourself over and over again to

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the point that you remember the rules.

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Because if you just memorize the rules, later on in life

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when you haven't done it for a couple of years, you might kind

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of forget the rules, and then you won't know how

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to get back to the rules.

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But it's up to you.

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I just hope you do understand why these rules work, and as

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long as you practice and you pay attention to the signs,

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you should have no problems with the Level 1 exercises.

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Have fun!

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