## What does it mean to raise a number to a power?

What if the power is whole number? An integer? A rational number? A real number?

For a specific example, how do we raise to the power? What about ?

Exponents begin in a very straightforward fashion: when the exponent is a positive integer, it means “repeated multiplication”.

**Positive integer powers**

**Definition.** If is a positive integer and is a positive real number, then

**Theorem.** If are positive integers and are real numbers, then:

The proof of each property is “obvious from the definition.” (Aside: are they really obvious? Try to explain one of these properties using the definition of *positive integer powers*).

If we choose a particular value for the number , such as , we get a function with domain equal to the positive integers.

To define for non-integer values of , we need to “fill in” between the points given by integer values (in math, this is called *interpolation*). However, there are lots of ways to fill in between points.

*Give some examples – discuss pros/cons.*

**The exponential function **

**Theorem. **If is a positive real number then there is a unique function satisfying properties 1, 2 and 3:

- is continuous
- For all positive integers , ( times)
- For all real numbers and , (that is, )

**Definition**. Let us refer to this unique function with the notation .

EXAMPLE: Let’s use this definition to figure out what should equal.

We know by property 3 that . Now the right side is , so it equals . The left side is multiplied by itself. Thus has the property that, when multiplied by itself, equals . We already have a name for this number, which is . Thus .

**Rational and negative exponents**

**Theorem.** If is a positive real number and is a positive rational number then .

**Theorem.** If is a positive real number and is a real number then .

EXAMPLE: Let’s use this definition to figure out what should equal.

We know how to rise to a rational power. If we choose rational numbers closer and closer to , the result will be closer and closer to . Strictly speaking, we are using the fact that this function is continuous (property 1) and taking a limit as of .

**The exponential function **

**Theorem. **If is a positive real number then there is a unique function satisfying properties 1, 2 and 3:

- is continuous
- For all positive integers , ( times)
- For all real numbers and , (that is, )

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