Welcome to the presentation on the Level 2 exponents.
In Level 2 exponents, the only thing we're going to add to
the mix now is the concept of a negative exponent.
So we learned already that 2 the third power, well, that
just equals 2 times 2 times 2.
Hopefully, by now, that's second nature to you,
and that equals 8.
Now I'm going to teach you what 2 to the negative
third power is.
I know a lot of you are going to say, oh, no, that's negative
8, but whenever you see exponents, you always have to
remind yourself, exponents are not multiplication.
I know there's that temptation to say, well, 2 times negative
3 is negative 6, so maybe 2 to the negative third power
is negative 8, but that's not the case.
And I'll explain in a future module why we use this
convention, but 2 to the negative third power, it
turns out, is equivalent to 1/2 to the third power.
So it turns out that the negative exponent, what it does
is it means to take the inverse of the base-- we'll call the
number 2 as the base-- and take that to the positive
version of the exponent.
And 1/2 to the third power?
Well, we already learned that's 1/2 times 1/2 times 1/2,
and that equals 1/8.
So we say 2 to the negative third power.
That didn't come out right.
2 to the negative third power is equal to 1/8.
Let's do another one.
Let's say 3 to the negative 2 power.
Once again, immediately when we see that negative in the
exponent, the easiest thing to do is just immediately take
the reciprocal of the base.
So we take 1/3.
And we raise that to the positive 2 power.
And that's easy enough.
1/3 squared, well, that's equal to 1/9.
Let's do some more problems.
What if I had 2/3 to the negative third power?
Once again, just to make it simple, whenever I see that
negative in the exponent, I want to get rid of it.
So I immediately take the reciprocal of the base.
The reciprocal of 2/3 is 3/2, and I raise that to the
positive third power.
So what changed between the left and the right side?
The 2/3 I flipped, and I turned the negative
3 into a positive 3.
And now this just becomes a Level 1 exponent.
This equals 3/2 times 3/2 times 3/2, and that equals 27/8.
So that's interesting.
2/3 to the negative 3 is equal to 27/8.
Let's do some more.
Let's do 4/7 to the negative 1.
Once again, we have a negative number in the exponent.
That's the same thing as taking the reciprocal of the base
and raising it to the positive exponent.
Well, 7/4 to the 1, any number to the first power
is just the same number.
It's equal to 7/4.
So when take it to the negative 1 power, all you're essentially
doing is getting the reciprocal of the number.
Let's do some more problems.
2 to the negative 5.
Once again, we take the reciprocal of 2, and we say
1/2, and now that can be raised to the fifth power, and that
equals 1/2 times 1/2 times 1/2 times 1/2 times 1/2,
and that equals 1/32.
Another way we could have viewed 2 to the negative fifth
is that, 2 to the negative fifth, we could have said that
equals 1 over 2 to the fifth.
And we know that 2 to the fifth is 32, so that would've
been the same thing.
Two ways to do it, pretty much just changing the order of when
you flip versus when you actually calculate
the exponent.
Let me do two or three more problems.
And after I write down each of these problems, you might just
want to pause it and see if you can do the problem yourself,
and then compare your answer to mine.
So let's say I had negative 4 to the negative third power.
Immediately, I like to get rid of the negative in the
exponent, and I know that that equals minus 1/4 to the third
power, and that equals minus 1/4 times minus 1/4
times minus 1/4.
The negative times a negative is a positive, but then we're
multiplying that times another negative, so we get a negative.
1 times 1 times 1 is 1.
4 times 4 is 16 times 4 is 64.
So it equals negative 1/64.
Let's do another problem.
Let me think of a good number.
8/9.
Let's make it negative.
Negative 8/9 to the negative second power.
Well, once again, that equals negative 9/8.
Notice, I immediately just took the reciprocal of the base to
the positive 2 power, and now that equals negative 9/8
times negative 9/8.
A negative times a negative is a positive, so we
get 9 times 9 is 81/64.
I think you get the point now.
The only new thing we've learned, really, is that when
you have a negative exponent, it's the same thing as taking
the reciprocal of the base and raising it to the
positive exponent.
Hopefully, that last statement didn't confuse you more, did
more good than damage, but I think you're ready to
try some problems now.
Have fun!