Handling exponents

Exponents behave nicely with respect to multiplication and division.

$$\begin{eqnarray*}(ab)^n&=&a^nb^n\ a^na^m&=&a^{n+m}\ \left(a^n\right)^m &=& a^{nm} \\ \left(\frac{a}{b}\right)^n&=& a^n b^{-n}\end{eqnarray*}$$

They do not behave nicely with respect to addition:

$$(a+b)^n\neq a^n+b^n$$

unless, of course, $n=1$.

Properties of logarithms follow from the properties of exponentials once we define

$$a^x=y \Leftrightarrow x = \log_a( y)$$

The equations above then tell us that

$$\begin{eqnarray*}\log_a(nm)&=&\log_a(n)+\log_a(m) \ \log_a(n^m)&=& m\log_a(n)\end{eqnarray*}$$

The rule

$$(a^n)^m=a^{nm}$$

tells us that

$$\left(a^{\frac1n}\right)^n=a$$

so

$$\sqrt[n]{a} = a^{\frac1n}$$

So for instance

$$\begin{eqnarray*}\sqrt{x} &=& x^{\frac12} \\ \sqrt[3]{x} &=& x^{\frac13}\\ \sqrt[3]{x^2} &=& x^{\frac23} \end{eqnarray*}$$