Precalculus Review

Rational and Irrational Exponents

Rational exponents. Let $n \ge 2$ be a positive integer, and let

be a positive real number. Then we can find a unique value

so that

. We call such value

the

-th root of

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

Compound interest. Suppose that represents the principal, and that is an interest rate. The amount after 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

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