Evaluating fractional exponents: negative unit-fraction

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.