# Music and Math Project

## 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!