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# Evaluating fractional exponents: negative unit-fraction

Video transcript

Let's do some slightly
more complicated fractional exponent examples. So we already know
that if I were to take 9 to the 1/2 power,
this is going to be equal to 3, and we know that because
3 times 3 is equal to 9. This is equivalent
to saying, what is the principal root of 9? Well, that is equal to 3. But what would happen if I took
9 to the negative 1/2 power? Now we have a negative
fractional exponent, and the key to
this is to just not get too worried or
intimidated by this, but just think about
it step by step. Just ignore for the second
that this is a fraction, and just look at
this negative first. Just breathe
slowly, and realize, OK, I got a negative exponent. That means that this is
just going to be 1 over 9 to the 1/2. That's what that
negative is a cue for. This is 1 over 9 to the 1/2,
and we know that 9 to the 1/2 is equal to 3. So this is just going
to be equal to 1/3. Let's take things a
little bit further. What would this evaluate to? And I encourage you to pause
the video after trying it, or pause the video to try it. Negative 27 to the
negative 1/3 power. So I encourage you
to pause the video and think about what
this would evaluate to. So remember, just
take a deep breath. You can always get rid of
this negative in the exponent by taking the reciprocal and
raising it to the positive. So this is going to be
equal to 1 over negative 27 to the positive 1/3 power. And I know what you're saying. Hey, I still can't
breathe easily. I have this negative number
to this fractional exponent. But this is just
saying what number, if I were to multiply
it three times-- so if I have that number, so
whatever the number this is, if I were to multiply it,
if I took three of them and I multiply them
together, if I multiplied 1 by that number three
times, what number would I have to use here
to get negative 27? Well, we already know
that 3 to the third, which is equal to
3 times 3 times 3, is equal to positive 27. So that's a pretty good clue. What would negative 3
to the third power be? Well, that's negative 3 times
negative 3 times negative 3, which is negative 3 times
negative 3 is positive 9. Times negative 3 is negative 27. So we've just found this
number, this question mark. Negative 3 times negative
3 times negative 3 is equal to negative 27. So negative 27 to the
1/3-- this part right over here-- is equal to negative 3. So this is going to be
equal to 1 over negative 3, which is the same
thing as negative 1/3.