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## Laws of exponents

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# Exponent rules part (composite problems)

## Video transcript

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. 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 eighteen 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 eighteenth 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!