Let's look at rational exponents. For m and n, natural numbers and b, any real
number, we have the following. The b to the m/n is either one of these. We can
either take the nth root of b first and then raise that the mth power or the other
way around. We can raise b to the mth power and then take the nth root of that.
However, note that b cannot be negative when n is even. Otherwise, we'd be taking
the even root of a negative number and we know we cannot do that. So, when are we
going to use the first method and when are we going to use the second? Let's see an
example. Let's say we wanted to compute eight to the two-thirds power. If we use
the first method this is equal to eight to the one-third power, whole thing squared,
and the cube root of eight is two, so this is two^2 which is equal to four.
Now, if we use the second method, we would first square the eight, so this is eight^2
to the one-third power, which is equal to 64 to the one-third power, and the cube
root of 64 is four. So, it doesn't matter, they're the same,
right? So, you can use either method. But in some cases, we'll want to use one
method over the other. For example, let's say, we wanted to compute 27 to the 4/3
power, we most definitely would want to do it this first way. Namely, this is equal
to 27 raised to the one-third power and then, that whole thing raised to the
fourth. And we know what the cube root of 27 is, right? That's three.
So, this is three^4 which is equal to 81. But if we tried to use the second method,
we would have to raise 27 to the fourth power and then take the cubed root of
that. It is much more difficult, right? Because, what is this? It would be more
challenging this way. So, it's sort of case specific on when you use which
method. Alright, what about rational exponents, that are negative? What we do
is we take one over b to the positive m divided by n. And then, we do what we did
before. For example, let's say, we wanted to compute nine to the -3/2 power. By
this, this is the same as one divided by nine to the +3/2 power. Now, the question
is, which method would we use to compute this denomina tor?
Would we compute nine^3 raised to the one-half power or nine to the one-half,
whole thing cubed. This here is much more promising, isn't it? Because, otherwise,
we would have to compute nine^3. Alright, so what is nine to the one-half
power? That is equal to three and three^3 = 27. Alright. So, this denominator here
then is 27. Therefore, our answer is one / 27. And this is how we work with rational
exponents. Thank you and we'll see you next time.