Zero exponent:

 

a to the zero power equals one where a cannot be zero

 

Negative exponents:

 

a to the negative n power equals 1 over a to the n power where a cannot be zero

 

Multiplication with common bases:

 

a to the x power times a to the y power equals a to the x plus y power

 

Division with common bases:

 

a to the x power divided by a to the y power equals a to the x minus y power where a cannot be zero

 

Raising an exponent to another exponent:

 

a to the x power raised to the y power equals a to the x times y power

 

Raising a product to an exponent:

 

the product ab raised to the n power equals a to the n power times b to the n power

 

Raising a quotient to an exponent:

 

the quotient a over b raised to the n power equals a to the n power divided by b to the n power where b cannot equal zero

 

Fractional exponents:

 

a raised to the fraction x over y equals the y root of a to the x power

 

nth root of a product:

 

the nth root of the product ab equals the nth root of a times the nth root of b

 

nth root of a quotient:

 

the nth root of the quotient a over b equals the nth root of a divided by the nth root of b

 

Taking the root of a root:

 

the mth root of the nth root of a equals the m times n root of a

 

Properties of nth roots:

 

the nth root of a to the n power equals a if n is odd

the nth root of a to the n power equals the absolute value of a if n is even