Laws of Exponents and More

Product of Powers

When multiplying two powers with the same base, add the exponents: a^m × a^n = a^(m+n).

Quotient of Powers

When dividing two powers with the same base, subtract the exponents: a^m ÷ a^n = a^(m−n).

Power of a Power

To raise a power to another power, multiply the exponents: (a^m)^n = a^(m×n).

Power of a Product

To raise a product to a power, raise each factor to that power: (ab)^n = a^n × b^n.

Power of a Quotient

To raise a quotient to a power, raise the numerator and denominator to that power: (a/b)^n = a^n / b^n.

Zero Exponent

Any non-zero number raised to the zero power is equal to 1: a^0 = 1 (where a ≠ 0).

Negative Exponent

Represents the reciprocal of the base raised to the opposite positive exponent: a^(-n) = 1/a^n.

Fractional Exponents

Denotes a root as well as a power: a^(m/n) = n√(a^m).

Identity Exponent

Any number to the power of one is itself: a^1 = a.

Factoring

Factoring is the process of expressing a polynomial as a product of its factors.

Common Factor

Identifying a greatest common factor for all terms in the expression and factoring it out.

Difference of Squares

Refers to the formula a^2 - b^2 = (a - b)(a + b).

Trinomials

Factoring quadratics into two binomials.

Factoring by Grouping

A method useful for polynomials with four or more terms, grouping pairs of terms to factor out common elements.

Perfect Square Trinomials

Recognizing patterns such as a^2 + 2ab + b^2 = (a + b)^2.

Sum and Difference of Cubes

Formulas: a^3 + b^3 = (a + b)(a^2 - ab + b^2) and a^3 - b^3 = (a - b)(a^2 + ab + b^2).