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).

Natural Numbers, which are expressed as counting numbers upward excluding 0, are noted by what letter?

N

Whole Numbers, which include natural numbers and zero, are expressed by what letter?

W

Rational Numbers, which are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero. They include integers, terminating decimals (like 1/2 = 0.5), and repeating decimals are expressed by what letter?

Q

Integers, which are include whole numbers, their negative counterparts, and zero (... -3, -2, -1, 0, 1, 2, 3, ...) are expressed by what letter?

Z

Irrational Numbers, which are real numbers that cannot be expressed as a fraction p/q, where p and q are integers. They are non-repeating, non-terminating decimals (like π or √2), are expressed by what letter?

I

What can’t polynomials have?

Negative Exponents, fractional exponents, variables in the denominator, radicals (including roots) of Variables, and infinite number of terms.

What do polynomials have?

Constants, variables, exponents, coefficents, and terms

Slope Intercept Form

y = mx + b

Point-Slope Form

y - y₁ = m(x - x₁)

Standard Form

Ax + By = C

Absolute Value

the distance a number is from zero on a number line, always resulting in a positive value or zero.

(a/b)^-1 equals what?

b/a

(a/b)^-m (m being smaller negative exponents besides 1) equals what?

b^m/a^m

In order to do proportions, you should what?

Cross, multiply, and then divide.