## Definitions

Terminology

We would call this “*x* raised to the *a* power”*x* is called the base*a* is called the exponent

Example:

is called “seven raised to the eleventh power”

the base is 7

the exponent is 11

Integer exponents

An integer exponent just means multiplying a number by itself.

Example:

Fraction exponents

An exponent which is the reciprocal of an integer means taking the root

Example:

Negative exponents

A negative exponent indicates a reciprocal

Example:

NOTICE: This means you don’t ever have to deal with fractions or roots ever again if you don’t want to! You can treat them all like exponents.

## Algebra rules

Combining exponents

When two numbers with the same base are multiplied, the exponents may be added

Example:

Exponents in ratios

When exponents in the numerator an denominator have the same base, their exponents may be subracted

NOTE: This is just a combination of the negative exponent rule with the multiplication rule!

Distributing exponents

If you have 2 layers of exponents, they may be combined

Example:

Negative bases

When the base is a negative number, the sign of the expression overall depends on whether the exponent is even or odd.

if *n* is an odd integer

if *n* is an even integer

Therefore:

if *n* is an odd integer

if *n* is an even integer

Examples:

A warning about addition

Exponents DO NOT distribute cleanly over addition

You either have to find a way to simplify the base first OR use the definition of the exponent to simplify things.

Examples: