 So a useful feature about mathematics is the same laws apply to everything. And in this case, whether the exponents are negative or fractional, the same rules still apply. So here's a flash review. If we multiply, we add the exponents. If we divide, we subtract. If we raise a power to a power, we multiply the exponents. If I'm raising a product to a power, the exponent applies to both. If I'm raising a quotient to a power, the exponent applies to both. Exponents of zero, as long as we're dealing with a positive number, are going to always be equal to one. The negative exponent is always one over, and a fractional exponent corresponds to a root. And the important idea is to take things one step at a time. And a useful idea to keep in mind is that the type of expression is determined by the last operation performed. So in this case, we have parentheses, which indicate that whatever's inside has to be taken care of first, and then we raise things to the power one-fifth. And what this means is that we have something raised to the power one-fifth. Now, since we're assuming all the variables are positive, we don't have to worry about absolute values. And here, since we're raising something to a power, the product rule applies. If I raise a product to a power, the power applies to the individual factors. And so I can rewrite this as... Now, if I take a closer look at these individual terms, it appears that the power of a power rule applies because these are all powers of a or b raised to some additional power. And so I multiply the two exponents to get my final answer. So here, the last thing we do is raise everything to the fourth power. So this is the fourth power of a quotient. And so that fourth power will apply to both numerator and denominator. Now, in the numerator, I have a power of a product. And so I can simplify that by applying the product to each of the individual factors. The denominator is a little bit more complicated. This is a cube root. And so first, I might want to rewrite the cube root as an exponential expression using my fractional exponent rule. And now my denominator is a power of a power. So I can multiply the two exponents. Now, I'll leave the five to the fourth as five to the fourth because that's just a number. And the rest of it is a quotient, a to the fourth over a to the fourth thirds. And so my rules of exponents say that if I have a quotient with the same base, I can simplify that by subtracting the exponents. So a to the fourth over a to the fourth thirds is a to the power of four minus four thirds, where we have to do a little bit of the arithmetic of fractions. And so our simplified form is a to the power of eight thirds. Now we do want to write our final answer without fractional or negative exponents, which means we have to take care of this fraction eight thirds. So we can use the power of a power rule to split that into the factors eight and one third. And since we have a fractional exponent, we can use the fractional exponent rule to read this as the cube root of a to the eighth. And that allows me to write my final answer without fractional or negative exponents. And no matter how complicated the expression is, we can always apply these rules one step at a time to simplify. So if some sadistic monster were to give you something like this... Wasn't that on our last exam? We can still do this by taking the process one step at a time. Now if the expression is particularly messy, it's useful to keep in mind that we are focusing on the very last thing that was done. So here the very last thing that was done was to take the cube root. So we could ignore the radicand for a moment. And since this is a cube root, it's the same as raising something to the one third power. And let's apply the kindergarten rule. Let's put things back where we found them. The radicand was this horrible frightening expression. Let's put that back where it was. Again, the last thing inside is raising something to the fifth power. And now I have a power raised to a power. So I can use the power of a power rule to simplify. And we apply the kindergarten rule, put things back where we found them. In the parentheses originally was this mess. And so we need to put it back. So we'll ignore the numerator and denominator for a moment and focus on the fact that it is a quotient. And so this is the power of a quotient. So to simplify the power of a quotient, we'll raise the numerator to the power and the denominator to the power. And then we'll put things back where we found them. Well, now we have a product to the five thirds. So we can raise each of the individual factors to the five thirds. And we have a power of a power. So I can multiply those exponents three times five thirds and two times five thirds, which means we have to do a little bit of fraction arithmetic. And so our numerator simplifies. We can also simplify the denominator because this is a root. We'll start by rewriting it as an exponent. It's a power of a power. So we can multiply the two exponents. And it's a power of a product. So we can apply the exponent to both of the individual factors to get a simplified form of the denominator. Now we have a fraction and it's helpful to remember that we can break a fraction apart into a product of fractions by splitting up the factors of the numerator and the factors of the denominator. So we'll rewrite this as a quotient of A's times a quotient of B's. And that's useful because now we have a quotient of powered expressions. And that means we can apply our quotient rule. A to the fifth over A to the five twelfths is A to the power of five minus five twelfths. And we have to do some fraction arithmetic to get 55 twelfths is our answer. And similarly, B to the ten-thirds over B to the five twelfths. Well, that's B to the ten-thirds minus five twelfths where we do some fraction arithmetic and get our final exponent, 35 twelfths. And finally, we might want to try to write this without fractional exponents. And so we might remember we can rewrite A to the 55 twelfths as A to the 55 to the one twelfth, which is a fractional exponent. And so that translates into a root. And likewise, we're B to the 35 twelfths.