Basic Combinations

Since the raising of a number n to a power p may be defined as multiplying n times itself p times, it follows that

The rule for raising a power to a power can also be deduced

Powers of Ten

For very large or very small numbers, it convenient to use the "scientific notation", i.e., to express the numbers in terms of a power of 10. Standard prefixes are used to denote certain powers of 10:

Radical Combinations

Euler Relationship

The trigonometric functions are related to a complex exponential by the Euler relationship

From these relationships the trig functions can be expressed in terms of the complex exponential:

Hyperbolic Functions

The hyperbolic functions can be defined in terms of exponentials.

There are closed-form expressions for the inverse hyperbolic functions:

An application of the hyperbolic tangent is in calculating the speed of ocean waves. The hyperbolic cosine and its inverse are encountered in fluid friction in the case of quadratic drag.