Common Core Standards: Math
2. Rewrite expressions using radical and rational exponents using the properties of exponents.
In the previous standard, we established some rules about fractional exponents. Here's the gist of those rules:
- We're allowed to have exponents that are fractions. It's really okay.
- The denominator of the fraction is the root.
- The numerator of the fraction is the power.
- It doesn't matter which we do first.
Those four little rules mean that it's easy to evaluate a lot of fractional exponents without the use of a calculator. Your students should know that math—real math with scary things like exponents—can be done without a calculator. Yes, using the calculator function on their phone counts as cheating. A pencil and an eraser, on the other hand, does not.
Now that we know these rules, we can go from radicals to exponents and vice versa. That'll help us turn some pretty ugly-duckling problems into gorgeous-swan answers. Or at least make the transition from duck to swan significantly less painful than ballet dancer to swan.
For example, let's rewrite without a radical and evaluate. The 3 outside the radical becomes the denominator. So what do we do for a numerator? The numerator is the power to which 64 is raised. Since no power is listed, we know it's a power of 1. (A number that doesn't have an exponent is to the first power.)
In exponential form, our answer is . To evaluate it, we just find the number that, when cubed is 64. That'd be 4.
What if we want to rewrite in radical form? Well, our numerator is 7 and our denominator is 2. We want the square root of 100 to the seventh power, or . (Where did the 2 go? For square roots, the 2 is assumed. Did we assume 2 much?)
If we raise 10 to the seventh power, we get the very long number 10,000,000. If only that was the number in your bank account, huh?