We can extend our connection between expressions with fractional exponents and expressions with roots to .

This means that we can interpret an expression like as or as .

Using the exponent rule can also be useful for writing roots in a simpler way. For example, it is true that because the first expression can be written as , which is equivalent to  .

The rule can also be helpful when interpreting certain situations. For example, let’s fill a chess board with rice by putting 1 grain of rice on the top left square, then doubling the amount of rice on each square as we go across the board. Will there ever be 1 million grains of rice on a square? If so, when?

We could write a function to represent the number of grains of rice on a square as , where represents the number of times the amount of rice on a square is doubled, so that there is 1 grain of rice to start, 2 grains after it has been doubled once, 4 grains after doubling twice, and so on. By graphing and we could figure out how many times it has been doubled to have 1 million grains of rice on a square. It turns out that the graphs intersect when . This means that when we double the rice on a square 20 times it will have more than 1 million grains of rice on it (and we are doubling the rice 43 more times for the last square of the board)!