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: 
