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- Welcome to Part 2 on the presentation on Level 1 exponent rules.
- So let's start off by reviewing the rules we've learned already.
- If I had 2 to the tenth times 2 to the fifth,
- we learned that since we're multiplying exponents with the same base, we can add the exponent,
- so this equals 2 to the fifteenth.
- We also learned that if it was 2 to the tenth over 2 to the fifth,
- we would actually subtract the exponents.
- So this would be 2 to the 10 minus 5,
- which equals 2 to the fifth.
- At the end of the last presentation,
- and I probably shouldn't have introduced it so fast,
- I introduced a new concept.
- What happens if I have 2 to the tenth to the fifth power?
- Well, let's think about what that means.
- When I raise something to the fifth power, that's just like saying
- 2 to the tenth times 2 to the tenth times 2 to the tenth
- times 2 to the tenth times 2 to the tenth, right?
- All I did is I took 2 to the tenth and I multiplied it by itself five times.
- That's the fifth power.
- Well, we know from this rule up here
- that we can add these exponents because they're all the same base.
- So if we add 10 plus 10 plus 10 plus 10 plus 10, what do we get?
- Right, we get 2 to the fiftieth power.
- So essentially, what did we do here?
- All we did is we multiplied 10 times 5 to get 50.
- So that's our third exponent rule, that when I raise an exponent to a power,
- and then I raise that whole expression to another power,
- I can multiply those two exponents.
- So let me give you another example.
- If I said 3 to the 7, and all of that to the negative 9,
- once again, all I do is I multiply the 7 and the negative 9,
- and I get 3 to the minus 63.
- So, you see, it works just as easily with negative numbers.
- So now, I'm going to teach you one final exponent property.
- If I have-- Let's say I have 2 times 9,
- and I raise that whole thing to the hundredth power.
- It turns out of this is equal to 2 to the hundredth power
- times 9 to the hundredth power.
- Now let's make sure that that makes sense.
- Let's do it with a smaller example.
- What if it was 4 times 5 to the third power?
- Well, that would just be equal to 4 times 5 times 4 times 5
- times 4 times 5, right, which is the same thing as 4 times 4
- times 4 times 5 times 5 times 5, right?
- I just switched the order in which I'm multiplying,
- which you can do with multiplication.
- Well, 4 times 4 times 4, well, that's just equal
- to 4 to the third.
- And 5 times 5 times 5 is equal to 5 to the third.
- Hope that gives you a good intuition of why this property here is true.
- And actually, when I had first learned exponent rules,
- I would always forget the rules,
- and I would always do this proof myself, or the other proofs.
- And a proof is just an explanation of why the rule works,
- just to make sure that I was doing it right.
- So given everything that we've learned to now--
- actually, let me review all of the rules again.
- If I have 2 to the seventh times 2 to the third,
- well, then I can add the exponents, 2 to the tenth.
- If I have 2 the seventh over 2 the third,
- well, here I subtract the exponents, and I get 2 to the fourth.
- If I have 2 to the seventh to the third power,
- well, here I multiplied the exponents.
- That gives you 2 to the 21.
- And if I had 2 times 7 to the third power,
- well, that equals 2 to the third times 7 to the third.
- Now, let's use all of these rules we've learned
- to actually try to do some, what I would call, composite problems
- that involve you using multiple rules at the same time.
- And a good composite problem was that problem that I had introduced you to at the end of that last seminar.
- Let's say I have 3 squared times 9 to the eighth power,
- and all of that I'm going to raise to the negative 2 power.
- So what can I do here?
- Well, 3 and 9 are two separate bases,
- but 9 can actually be expressed as an exponent of 3, right?
- 9 is the same thing as 3 squared.
- So let's rewrite 9 like that.
- That's equivalent to 3 squared times--
- 9 is the same thing as 3 squared to the eighth power,
- and then all of that to the negative 2 power, right?
- All I did is I replaced 9 with 3 squared,
- because we know 3 times 3 is 9.
- Well, now we can use the multiplication rule on this to simplify it.
- So this is equal to 3 squared times 3 to the 2 times 8,
- which is 16.
- And all of that to the negative 2.
- Now, we can use the first rule.
- We have the same base, so we can add the exponents, and we're multiplying them,
- so that equals 3 to the eighteenth power, right?
- 2 plus 16, and all that to the negative 2.
- And now we're almost done.
- We can once again use this multiplication rule,
- and we could say 3-- this is equal to 3 to the eighteen times negative 2,
- so that's 3 to the minus 36.
- So this problem might have seemed pretty daunting at first,
- but there aren't that many rules,
- and all you have to do is keep seeing, oh, wow, that little part of the problem, I can simplify it.
- Then you simplify it,
- and you'll see that you can keep using rules until you get to a much simpler answer.
- And actually the Level 1 problems don't even involve problems this difficult.
- This'll be more on the exponent rules, Level 2.
- But I think at this point you're ready to try the problems.
- I'm kind of divided whether I want you to memorize the rules
- because I think it's better to almost forget the rules
- and have to prove it to yourself over and over again to the point that you remember the rules.
- Because if you just memorize the rules,
- later on in life, when you haven't done it for a couple of years,
- you might kind of forget the rules,
- and then you won't know how to get back to the rules.
- But it's up to you.
- I just hope you do understand why these rules work,
- and as, long as you practice and you pay attention to the signs,
- you should have no problems with the Level 1 exercises.
- Have fun!