Exponential functions > Real exponents
12345Real exponents

## Theory

With exponential growth you have to multiply by the same number every unit of time. This number is called the growth rate for that time unit. If $g$ is the growth rate then: $g>0$.
To be able to use negative and/or rational exponents we need to agree about the following:

• negative exponents: ${g}^{-n}=\frac{1}{{g}^{n}}$

• rational exponents: ${g}^{\frac{1}{n}}=\sqrt[n]{g}$

These are valid for $g>0$ and positive integer $n$.

Both completely fit in with the computational rules for powers, such as:
${g}^{-n}=\frac{{g}^{0}}{{g}^{n}}=\frac{1}{{g}^{n}}$

This shows that a power ${g}^{a}$ for $g>0$ is meaningful if the exponent $a$ is either a positive number, a zero, a negative or a rational number.

The exponent $a$ can be any real number in fact.
And that is why the graph of an exponential function can be drawn as smooth curves.

Here you see the graph of $B=6\cdot {2}^{t}$.