When a is a nonzero Real number and n and m are Whole numbers, then

a³ expands to a • a • a

aⁿ • a^m = a^{n + m}    The Product Rule: when multipling two numbers with the same base, add the exponents

aⁿ / a^m = a^{n - m}  The Quotient Rule: when dividing two numbers with the same base, subtract the exponent of the denominator (divisor) from the exponent of the numerator (dividend).

(aⁿ)^m = a^nm        The Power Rule: when raising an expponetiated number to a power, multiply the exponent times the power.

(x + a)²  expands to   (x + a)(x + a)   and simplifies to   x² + 2ax + a²

(x – a)²   expands to   (x – a)(x – a)  and simplifies to   x² – 2ax + a²

When squaring a binomial, the product will ALWAYS be a trinomial!

Conjugate binomials    Difference of Squares
   (x + a)(x – a)                  x² – a²

A positive exponent means that we are multiplying by the base.
Ex: 2³ = 2 • 2 • 2 = 8

A negative exponent means that we are dividing by the base.
Ex: 2^–3 = 1/2 • 1/2 • 1/2 = 1/8 

To change the sign of the exponent, invert (flip) the base.
Ex: 2^–3 = (1/2)³ = 1/8

When a base has a negative exponent, it needs to move to the other side of the fraction bar.
Ex: x^-3 / y-² = y² / x³