Exponentials with positive integer exponents

The expression $a^n$ means $\underbrace{a \cdot a \cdot a \cdot\ldots \cdot a}_{n \text{ times}}$, where we multiply $a$ by itself $n>0$ times. Note that $1^n=1$ and $0^n=0$. The three laws of exponents are:

  • $\displaystyle{a^b \cdot a^c = a^{b+c}}$

  • $\displaystyle{\frac{a^b}{a^c} = a^{b-c}}$, and

  • $\displaystyle{(a^b)^c = a^{bc}}$.

These laws are explained in the following video:



 

Viewed as a function of $n$, the function $2^n$ grows very quickly after a while, much faster than powers of $n$.

$$\begin{matrix} n & & n^2 & & n^3 & & 2^n \\ \hline \\ 1& & 1 & & 1 & & 2 \\ 2& & 4 & & 8 & & 4 \\ 3 & & 9 & & 27 & & 8 \\ 4 & & 16 & & 64 & & 16 \\ 5 & & 25 & & 125 & & 32 \\ 10 & & 100 & & 1000 & & 1024 \\ 20 & & 400 & & 8000 & & 1,048,576 \\ 30 & & 900 & & 27,000 & & 1,073,741,824 \\ \ldots&&\ldots&&\ldots&&\ldots \end{matrix}$$

The same thing goes for any function $a^n$ with $a > 1$. On the other hand, if $a < 1$, then $a^n$ shrinks rapidly:

$$\left(\frac{1}{2}\right)^{10} = \frac{1}{1024}, \qquad \left(\frac{1}{2}\right)^{20} = \frac{1}{1,048,576}, \qquad \left(\frac{1}{2}\right)^{30} = \frac{1}{1,073,741,824}.$$