Power functions > Powers
12345Powers

## Theory

Take a look at the applet: Power Functions

If $y$ is directly proportional to a power of $x$, so $y=c\cdot {x}^{p}$, then this is called a power function. The constant $c$ is called the proportionality constant.

You can look at a few examples of power functions here. In these functions, $p$ is always positive or $0$ and $c=1$.

There are two ways to reverse the calculation in a power function $y={x}^{p}$ (thus with $c=1$ ):

• $x=\sqrt[p]{y}$

• $x={y}^{\frac{1}{p}}$

Depending on the value of $p$ you can get one or two values for x.
If the proportionality factor has a value other than $1$, then you have to start with dividing by $c$. From there you can either apply the root of power $p$, or use the inverse power.

The rules for working with powers (see: "Exponential functions" ) are valid here, too!

For every $x$ and any real numbers $a$ en $b$ the following properties of powers and exponents apply:

• ${x}^{0}=1$

• ${x}^{-a}=\frac{1}{{x}^{a}}$ as long as $x=!0$

• ${x}^{\frac{1}{a}}=\sqrt[a]{x}$ as long as $x\ge 0$ en $a>0$.

• ${x}^{a+b}={x}^{a}\cdot {x}^{b}$

• ${x}^{a-b}=\frac{{x}^{a}}{{x}^{b}}$ as long as $x=!0$

• ${\left({x}^{a}\right)}^{b}={x}^{a\cdot b}$