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 2 to the \frac{1}{2} power? What about 2^\pi?

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

Positive integer powers

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

    \[a^n = a\cdot a \cdot a \dots a \text{($n$ many $a$'s multiplied together)}\]

Theorem. If n,m are positive integers and a, b are real numbers, then:

  1. (a^n)(a^m) = a^{n+m}
  2. (a^n)^m = a^{nm}
  3. (ab)^n = a^n b^n

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 a, such as a=2, we get a function C(n)=2^n with domain equal to the positive integers.

Demonstration (Desmos).

To define 2^x for non-integer values of x, 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 a^x

Theorem. If a is a positive real number then there is a unique function f(x):\mathbb{R}\rightarrow\mathBB{R} satisfying properties 1, 2 and 3:

  1. f(x) is continuous
  2. For all positive integers n, f(n)=2^n = n\cdot n \cdot n \dots n (n times)
  3. For all real numbers s and t, f(s)\cdot f(t) = f(s+t) (that is, a^s a^t = a^{x+t})

Definition. Let us refer to this unique function with the notation a^x.

EXAMPLE: Let’s use this definition to figure out what 2^\frac{1}{2} should equal.

We know by property 3 that (2^\frac{1}{2}) (2^\frac{1}{2}) = 2^\left(\frac{1}{2}+\frac{1}{2}\right). Now the right side is 2^1, so it equals 2. The left side is 2^\frac{1}{2} multiplied by itself. Thus 2^\frac{1}{2} has the property that, when multiplied by itself, equals 2. We already have a name for this number, which is \sqrt{2}. Thus 2^\frac{1}{2} = \sqrt{2}.

Rational and negative exponents

Theorem. If a is a positive real number and \frac{m}{n} is a positive rational number then a^\frac{m}{n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m.

Theorem. If a is a positive real number and s is a real number then a^{-s}=\frac{1}{s}.

EXAMPLE: Let’s use this definition to figure out what 2^\pi should equal.

We know how to rise 2 to a rational power. If we choose rational numbers closer and closer to \pi, the result will be closer and closer to 2^\pi. Strictly speaking, we are using the fact that this function is continuous (property 1) and taking a limit as x\rightarrow \pi of 2^x.

The exponential function a^x

Theorem. If a is a positive real number then there is a unique function f(x):\mathbb{R}\rightarrow\mathBB{R} satisfying properties 1, 2 and 3:

  1. f(x) is continuous
  2. For all positive integers n, f(n)=a^n = a\cdot a \cdot a \dots a (n times)
  3. For all real numbers s and t, f(s)\cdot f(t) = f(s+t) (that is, a^s a^t = a^{x+t})