## Powers

### Examples

When we write
\[
2^4,
\]
this means \(2\) raised to the power of \(4\). It stands for \(2 \times 2 \times 2 \times 2\). In this case, \(2\) is called the base and \(4\) is the exponent.

The exponent does not have to be an integer. For instance, we can calculate \( 3^{\frac{1}{7}} \). Actually, it is the number such that if you raise it to the power \(7\), you get \(3\). In other words,
\[
3^{\frac{1}{7}} = \sqrt[7]{3}.
\]

If we want to calculate \(4^{\frac{5}{6}}\), we can first calculate \(4^{\frac{1}{6}}\), which is the same as \(\sqrt[6]{4}\), and then raise the answer to the power of \(5\),
\[
4^{\frac{5}{6}} = (4^{\frac{1}{6}})^5.
\]

### Explanation with symbols

We gave some examples above, but to be more precise, let us use some symbols from now on. Once more, if \(n\) is a positive integer then
\[
b^n = b \times \cdots \times b,
\]
where the factor \(b\) appears \(n\) times on the right hand side of the equation. As before \(b\) is called the base and \(n\) is the exponent. Next,
\[
b^{\frac{1}{n}} = \sqrt[n]{b},
\]
and
\[
b^{-n} = \frac{1}{b^n}.
\]
Also remember that for every base \(b\),
\[
b^0 = 1.
\]

### Rules for calculation

If \(b\) and \(c\) are positive numbers, and \(m\) and \(n\) are real numbers, the following rules always hold
\begin{align*}
b^m \cdot b^n &= b^{m+n}, \\
(b^m)^n &= b^{m\cdot n}, \\
(b\cdot c)^n &= b^n \cdot c^n.
\end{align*}

### Important remark about notation

It is important to remember that when we write \(b^{m^n}\), it means \(b^{(m^n)}\), that is: first take \(m\) to the power of \(n\) and use the result as the exponent to which you raise \(b\). The order matters!