指數法則 2 (英)

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指數法則 2 (英) : 再介紹兩個指數法則

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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!

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