Evaluating Rational Exponents - Concept
Rational exponents indicate two properties: the numerator is the base's power and the denominator is the power of the root. When evaluating rational exponents it is often helpful to break apart the exponent into its two parts: the power and the root. To decide if it is easier to perform the root first or the exponent first, see if there exists a whole number root of the base; if not, we perform the exponent operation first.
Evaluating an expression with rational exponents, so the main trick for evaluating something with rational exponents is to remember that what our fraction really is telling us is the power and the root that we're concerned with. Okay, so when I evaluate these the first thing I always do is rewrite them breaking up the fraction into the 2 components; the power part and the root part.
There's two ways of doing that, we could either say we could do our root first so remember when we take exponents to exponents we multiply so this ends up being the same thing here or we could do the exponent first and then the root okay either way this works is when we distribute this through we would end up with the same statement. Order of operations always tells us to start inside the parentheses and work out, so what we're concerned with here is the 8 to the one third remember what that means this really just the cube root of 8, cube root of 8 is just 2, so what we have here is 2 squared which is 4 okay. Doing it the other way, 8 squared 64, 64 to the one third again the one third is just the same thing as the cube root, cube root of 64 is just 4. Obviously we don't have to do this twice I'm just showing you there are 2 options.
In general I will always go for this one first do the root first and why do that is because our numbers are getting smaller okay? We take the root of something we're going to get smaller numbers. I like dealing with smaller numbers better than larger numbers. Here what we're doing is making our numbers larger okay? Imagine this was to the fourth power, I don't know what 8 to the fourth is so therefore I'm going to get huge numbers and it's going to hard to deal with so if we do our root first we end up getting our numbers smaller can be easier to manipulate. But the general principle is just breaking up our fraction into its power component and square root its root component and then simplifying as we go.