Rules for Rational Exponents - Problem 3

Simplifying an expression with rational exponents. So behind me I have a pretty ugly square root and we're going to try to simplify this up as much as possible. Whenever I see something like this where we're dealing with combinations of square root and exponents, I always rewrite my square roots as exponents. I am not really comfortable dealing with combining square roots, I am comfortable combining exponents, so what I'm going to do is rewrite everything remembering that square root is actually everything to the 1/2, so then the cube root is everything to a 1/3.

So just rewriting this, what we end up with is x to the fifth, y to the -2, x to the negative fourth, y to the 1/3, all to the 1/2. So where we can go from here is to combine like terms or distribute this this 1/2 in, for this example what I'm going to do is combine all my like terms, so focus on inside the parenthesis first. What this -4 does is bring the x to the fourth to the top and what this -2 does is bring this y² down, so what we have then is x to the fifth, x to the fourth, y to the 1/3, y², all to the 1/2.

Now we're going to multiply bases, we just have to add our exponents, so we're going to end up with x to the ninth in the top because we just add 4 and 5 and then we're adding 1/3 and 2, 2 is the same thing as 6/3, so we're going to end up with 7/3 as a the power of y. So let's come over and write that up.

We end up with x to the ninth over y to the, I said 7/3 didn't I? Try it again y to the 7/3 all to the 1/2. So now we're dealing with a power to a power situation, basically means we're going to need to multiply our exponents, so 9 times 1/2 is just going to be x to the 9/2 and y to the 7/3 to the 1/2 again multiplying leaves us with y to the 7/6.

This typically is going to be a fine answer. If you rewrite it as radicals you could do that. In this case we know that power over root and this is going to be the square root of x to the ninth, power over root so this is the sixth root of y to the seventh. You don't have to necessarily do that unless your teacher is asking for a radical form, but these are pretty much exactly the same thing one in an exponential form, one in radical form.

So by simplifying using our laws of exponents simplifying an expression with rational exponents is really no different than simplifying something with normal whole number exponents.