# Rules of Exponents

## 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: