This week in precalc 11, I learned how to evaluate an exponent that is a fraction. In the textbook, this is known as “writing powers with rational exponents”. Since all fractions are rational numbers, it is possible to make fractions into exponents on numbers. Here is an example:
Below are some other examples. As you can see, the fraction can be positive or negative:
To solve a fractional exponent, you change the base number from being a power, into a radicand within a radical. In my class I learned the saying “flower power, because the root is always on the bottom”. This is a helpful way to remember that the denominator (or bottom number) in the fractional exponent becomes the index in the radical.
This example below demonstrates the steps of changing a number with a positive fractional exponent into a radical. The numerator (top of fraction), can be placed as the exponent either on just the radicand, or the entire radical. I prefer to do it whichever way makes the numbers easier to solve. After this is done, the radical can be further evaluated.
This example shows the same steps, with a negative exponent. There are different ways to do this, but I chose the one I like best: