Simplify the expression with NO negative exponents. 3 − 4 5 − 2 \frac{3^{-4}}{5^{-2}}

So, in some of your problems, you may actually have a fraction in which both the numerator and the denominator both have negative exponents. Let's take a look at our practice problem here. We have three to the negative fourth power divided by five to the negative two power. So, we know that from our rules of negative exponents, our numerator should flip to the bottom, so it should flip to the denominator with a positive exponent, and then the reverse is true for the denominator it should flip up to the top with a positive exponent. What happens here is that when you have situations where both expressions in the top and bottom have a negative, they kind of just trade places. So, I don't have to do this thing where we have to kind of write, both fractions separately like multiplied together. You can actually just write this as a single fraction because we know the numerator is going to be five squared because the three to the negative fourth power will actually drop to the bottom. So, what happens here is this will drop to the bottom of the denominator, this will be three to the fourth power. So, see how basically they just straight traded places and then in both situations the exponent just became positive. Alright, so again, you can kind of do this to kind of shortcut some of these things, but then what happens? So, we just have to simplify this. So, these are all numbers these are not variables. We have to kind of simplify this. Five squared is just five times itself, which is 25, and then three to the fourth power, if you don't know off the top of your head, you can go just go ahead and sort of rearrange this. This is the same thing as three times three times three times three. Three times three is nine, so you basically have nine times itself twice. In other words, this is just 81. Now, can you simplify 25 over 81? Actually, no, you can't because the only thing that's common in them is one. The only roots of five are five and it's and one, and, for 81, you don't have anything that can go into 25. Alright? So, this is the most simplified that you can possibly write this. Again, the key idea here is that when both exponents are negative, you can actually sort of just flip their places, in the fraction and then rewrite them with positive exponents. Thanks for watching. Let me know if you have any questions.