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