Rational and irrational exponents

Rational exponents. Let $ n \ge 2$ be a positive integer, and let $a > 0$ be a positive real number. Then we can find a unique value $ b > 0$ so that $ b^n = a$. We call such value $b$ the $n$-th root of $a$, and write `` $ b = \sqrt[n]{a}$.'' Furthermore, for any rational number  $ \frac{m}{n}$ we can define the rational exponents  $ a^{\frac{m}{n}}$ by

$\displaystyle a^{\frac{1}{n}} = \sqrt[n]{a}, \quad
a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m},$    and $\displaystyle \quad
a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}.

Irrational exponents. Let $x$ be an irrational number. Then, for a rational number  $ \frac{m}{n}$ arbitrarily close to $x$ we can find a unique value $ b > 0$ so that the rational exponent  $ a^{\frac{m}{n}}$ becomes arbitrarily close to $b$. We call such value $b$ the irrational exponent $ a^x$.

Compound interest. Suppose that $ P$ represents the principal, and that $r$ is an interest rate. The amount $ A$ after $ k$ interest period is expressed by

$\displaystyle A = P(1 + r)^k

The number $ e$. We can find a unique value  $ e \approx 2.71828\ldots$ so that

$\displaystyle \left(1 + \frac{1}{n}\right)^n \to e \approx 2.71828\ldots$   $\displaystyle \mbox{ as $n \to \infty$ }$ (1)

Department of Mathematics
Last modified: 2005-09-29