Assigning a value to 00 The rules of exponents tell us that a0 = 1 if a is any number di...

Assigning a value to 0⁰ The rules of exponents tell us that a⁰ = 1 if a is any number different from zero. They also tell us that 0ⁿ = 0 if n is any positive number.

If we tried to extend these rules to include the case 0⁰, we would get conflicting results. The first rule would say 0⁰ = 1, whereas the second would say 0⁰ = 0.

We are not dealing with a question of right or wrong here. Neither rule applies as it stands, so there is no contradiction. We could, in fact, define 0⁰ to have any value we wanted as long as we could persuade others to agree.

What value would you like 0⁰ to have? Here is an example that might help you to decide. (See Exercise 2 below for another example.)

a. Calculate x^x for x = 0.01, 0.001, and so on as far as your calculator can go. Record the values you get. What pattern do you see?

b. Graph the function y = x^x for 0 < x ≤ 1. Even though the function is not defined for x ≤ 0, the graph will approach the y-axis from the right. Toward what y-value does it seem to be headed? Zoom in to further support your idea.

2. A reason you might want 0⁰ to be something other than 0 or 1 As the number x increases through positive values, the numbers 1/x and 1/(ln x) both approach zero. What happens to the number

as x increases? Here are two ways to find out.

a. Evaluate ƒ for x = 10, 100, 1000, and so on as far as your calculator can reasonably go. What pattern do you see?

b. Graph ƒ in a variety of graphing windows, including windows that contain the origin. What do you see? Trace the y-values along the graph. What do you find?