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# Exponent rules part 1

Introduction to exponent rules. Created by Sal Khan.

Video transcript

Welcome to the presentation
on level one exponent rules. Let's get started
with some problems. So if I were to ask you what 2
-- that's a little fatter than I wanted it to be, but let's
keep it fat so it doesn't look strange -- 2 the third times --
and dot is another way of saying times -- if I were to
ask you what 2 to the third times 2 to the fifth is, how
would you figure that out? Actually, let me use a skinnier
pen because that does look bad. So, 2 to the third
times 2 to the fifth. Well there's one way that I
think you do know how to do it. You could figure out that
2 to the third is 8, and that 2 to the fifth is 32. And then you could
multiply them. And 8 times 32 is 240,
plus it's 256, right? You could do it that way. That's reasonable because it's
not that hard to figure out 2 to the third is and what
2 to the fifth is. But if those were much larger
numbers this method might become a little difficult. So I'm going to show you using
exponent rules you can actually multiply exponentials or
exponent numbers without actually having to do as much
arithmetic or actually you could handle numbers much
larger than your normal math skills might allow you to. So let's just think what
2 to the third times 2 to the fifth means. 2 to the third is 2
times 2 times 2, right? And we're multiplying that
times 2 to the fifth. And that's 2 times 2
times 2 times 2 times 2. So what do we have here? We have 2 times 2 times
2, times 2 times 2 times 2 times 2 times 2. Really all we're doing is we're
multiplying 2 how many times? Well, one, two, three, four,
five, six, seven, eight. So that's the same thing
as 2 to the eighth. Interesting. 3 plus 5 is equal to 8. And that makes sense because 2
to the 3 is 2 multiplying by itself three times, to the
fifth is 2 multiplying by itself five times, and then
we're multiplying the two, so we're going to multiply
2 eight times. I hope I achieved my goal
of confusing you just now. Let's do another one. If I said 7 squared
times 7 to the fourth. That's a 4. Well, that equals 7 times 7,
right, that's 7 squared, times and now let's
do 7 to the fourth. 7 times 7 times 7 times 7. Well now we're multiplying
7 by itself six times, so that equal 7 to the sixth. So in general, whenever I'm
multiplying exponents of the same base, that's key, I can
just add the exponents. So 7 to the hundredth power
times 7 to the fiftieth power, and notice this
is an example now. It would be very hard without
a computer to figure out what 7 to the hundredth power is. And likewise, very hard without
a computer to figure out what 7 to the fiftieth power is. But we could say that this is
equal to 7 to the 100 plus 50, which is equal to 7 to the 150. Now I just want to give you a
little bit of warning, make sure that you're multiplying. Because if I had 7 to the 100
plus 7 to the 50, there's actually very little
I could do here. I couldn't simplify
this number. But I'll throw out one to you. If I had 2 to the 8 times
2 to the 20, we know we can add these exponents. So that gives you 2
to the 28, right? What if I had 2 to the
8 plus 2 to the 8? This is a bit of a
trick question. Well I just said if
we're adding, we can't really do anything. We can't really simplify it. But there's a little trick
here that we actually have two 2 to the 8, right? There's 2 to the 8 times
1, 2 to the 8 times 2. So this is the same thing as 2
times 2 to the 8, isn't it? 2 times 2 to the 8. That's just 2 to
the 8 plus itself. And 2 times to the 8, well
that's the same thing as 2 to the first times 2 to the 8. And 2 to the first times 2 to
the 8 by the same rule we just did is equal to 2 to the 9. So I thought I would just
throw that out to you. And it works even with
negative exponents. If I were to say 5 to the
negative 100 times 3 to the, say, 100 -- oh sorry, times
5 -- this has to be a 5. I don't know what my
brain was doing. 5 to the negative 100 times
5 to the 102, that would equal 5 squared, right? I just take minus 100 plus 102. This is a 5. I'm sorry for that
brain malfunction. And of course, that equals 25. So that's the first
exponent rule. Now I'm going to show you
another one, and it kind of leads from the same thing. If I were to ask you what 2 to
the 9 over 2 to the 10 equals, that looks like that could
be a little confusing. But it actually turns out to be
the same rule, because what's another way of writing this? Well, we know that this is also
the same thing as 2 to the 9 times 1 over 2 to
the 10, right? And we know 1 over 2 to the 10. Well, you could re-write right
this as 2 the 9 times 2 to the negative 10, right? All I did is I took 1 over 2 to
the 10 and I flipped it and I made the exponent negative. And I think you know
that already from level two exponents. And now, once again, we can
just add the exponents. 9 plus negative 10 equals 2 to
the negative 1, or we could say that equals 1/2, right? So it's an interesting
thing here. Whatever is the bottom
exponent, you could put it in the numerator like we did here,
but turn it into a negative. So that leads us to the second
exponent rule, simplification is we could just say that this
equals 2 to the 9 minus 10, which equals 2 to
the negative 1. Let's do another
problem like that. If I said 10 to the 200 over
10 to the 50, well that just equals 10 to the 200
minus 50, which is 150. Likewise, if I had 7 to the
fortieth power over 7 to the negative fifth power,
this will equal 7 to the fortieth minus negative 5. So it equals 7 to
the forty-fifth. Now I want you to think about
that, does that make sense? Well, we could have re-written
this equation as 7 to the fortieth times 7 to
the fifth, right? We could have taken this 1 over
7 to the negative 5 and turn it into 7 to the fifth, and that
would also just be 7 to the forty-five. So the second exponent rule I
just taught you actually is no different than that first one. If the exponent is in the
denominator, and of course, it has to be the same base and
you're dividing, you subtract it from the exponent
in the numerator. If they're both in the
numerator, as in this case, 7 to the fortieth times 7 to the
fifth -- actually there's no numerator, but they're
essentially multiplying by each other, and of course, you have
to have the same base. Then you add the exponents. I'm going to add one variation
of this, and actually this is the same thing but it's a
little bit of a trick question. What is 2 to the 9
times 4 to the 100? Actually, maybe I shouldn't
teach this to you, you have to wait until I teach
you the next rule. But I'll give you
a little hint. This is the same thing as 2 the
9 times 2 squared to the 100. And the rule I'm going to teach
you now is that when you have something to an exponent and
then that number raised to an exponent, you actually
multiply these two exponents. So this would be 2 the
9 times 2 to the 200. And by that first rule
we learned, this would be 2 to the 209. Now in the next module
I'm going to cover this in more detail. I think I might have
just confused you. But watch the next video and
then after the next video I think you're going to be ready
to do level one exponent rules. Have fun.