 We can further simplify rational exponents by using a couple of important ideas. First, we have the power of a power, provided that both sides exist. a to power m to power n is the same as a to the power of the product m times n. It's also helpful to remember the rule for the product of fractions. And in particular, a over b is the same as a times 1 over b. And finally, commutativity is also useful for a and b real numbers. a times b is the same as b times a. So for example, let's say I want to simplify 125 to power two-thirds. So since this is a fraction two-thirds, it's useful to remember that we have a rule for the product of fractions. a over b can be rewritten as a times 1 over b. So this exponent two-thirds can be rewritten as two times one-third. Now a useful thing to remember here is that if you take the root first, you'll generally get a number that's easier to work with. So I don't really want to find 125 squared. I would much rather take the one-third power, the cube root. Fortunately, we're allowed to do that. For a and b real numbers, a times b is the same as b times a. So this power two times one-third can be rewritten as one-third times two. Now I can use the power of a power rule. I have a product of exponents one-third times two, and I can split that up 125 to power one-third. The whole mess raised to power two. Remember that fractal exponent one-third is the same as a cube root of 125. So we'll rewrite it. And we can actually take the cube root of 125. We're still squaring the result so we get five squared, which is 25.