CLEP College Algebra cheat sheet

Laws of exponents: a^m·a^n = a^(m+n), a^m/a^n = a^(m−n), (a^m)^n = a^(mn), (ab)^n = a^n·b^n, a^(−n) = 1/a^n, a^0 = 1 for a ≠ 0.

Fractional exponents: a^(m/n) = the n-th root of a^m = (n-th root of a)^m; the n-th root of a^n equals |a| when n is even.

Perfect squares to memorize: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225; perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

Difference of squares: a² − b² = (a − b)(a + b). Perfect square trinomials: a² ± 2ab + b² = (a ± b)².

Sum and difference of cubes: a³ + b³ = (a + b)(a² − ab + b²); a³ − b³ = (a − b)(a² + ab + b²).

FOIL/monomial distribution: (a + b)(c + d) = ac + ad + bc + bd; each term of one factor multiplies every term of the other.

Rationalizing: multiply numerator and denominator by the conjugate (a − √b) to remove a radical from a denominator such as 1/(a + √b).

Simplification rule: cancel only factors common to the entire numerator and entire denominator, never individual terms; (x + 2)/2 is not x + 1.

Linear equation ax + b = c solves to x = (c − b)/a for a ≠ 0; a linear inequality reverses its direction when both sides are multiplied or divided by a negative number.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); discriminant b² − 4ac > 0 gives two real roots, = 0 gives one repeated root, < 0 gives no real roots.

Zero-product property: if (x − r)(x − s) = 0 then x = r or x = s; factoring requires the equation set equal to zero first.

Absolute value: |u| = k with k > 0 gives u = k or u = −k; |u| < k gives −k < u < k; |u| > k gives u < −k or u > k.

Compound inequalities: 'and' inequalities intersect solution sets (shaded overlap), 'or' inequalities union them; express answers in interval notation with brackets for included endpoints.

Systems of two linear equations: substitution isolates one variable; elimination adds multiples of equations to cancel a variable; parallel lines give no solution, identical lines give infinitely many.

Rational equations: multiply by the least common denominator, then reject any candidate that makes an original denominator zero (extraneous solution).

Exponential equations: rewrite both sides with the same base so a^u = a^v implies u = v; otherwise take a logarithm of both sides.

Definition: a function assigns exactly one output to each input; the vertical line test confirms a graph is a function, and the horizontal line test confirms it is one-to-one.

Domain restrictions: exclude values making a denominator zero and values making an even-index radical's radicand negative; polynomial and exponential domains are all reals.

Function arithmetic and composition: (f + g)(x) = f(x) + g(x); (f ∘ g)(x) = f(g(x)), evaluated inside-out, and generally f(g(x)) ≠ g(f(x)).

Inverse functions: f⁻¹ exists only if f is one-to-one; find it by swapping x and y and solving; the graphs reflect across y = x, and f(f⁻¹(x)) = x.

Transformations of y = f(x): f(x) + k shifts up k; f(x − h) shifts right h; −f(x) reflects over the x-axis; f(−x) reflects over the y-axis; a·f(x) stretches vertically by |a|.

Quadratic form: vertex of y = ax² + bx + c is at x = −b/(2a); vertex form y = a(x − h)² + k has vertex (h, k); a > 0 opens upward, a < 0 downward.

Exponential growth/decay: y = a·b^x with b > 1 growth and 0 < b < 1 decay; compound interest A = P(1 + r/n)^(nt).

Even and odd functions: f is even if f(−x) = f(x) (y-axis symmetry), odd if f(−x) = −f(x) (origin symmetry); rate of change is (f(b) − f(a))/(b − a).

Set hierarchy: natural numbers ⊂ integers ⊂ rationals ⊂ real numbers; irrationals are real but not rational, and rationals are exactly numbers expressible as p/q with integers p, q and q ≠ 0.

A rational number's decimal expansion either terminates or repeats; a nonterminating, nonrepeating decimal is irrational (for example √2, π, e).

Closure: integers are closed under addition, subtraction, and multiplication but not division; rationals are closed under all four operations except division by zero.

Order of operations: grouping symbols, then exponents and radicals, then multiplication and division left to right, then addition and subtraction left to right.

Properties: commutative a + b = b + a and ab = ba; associative regrouping; distributive a(b + c) = ab + ac; additive inverse −a and multiplicative inverse 1/a.

Absolute value: |a| is the distance from a to zero on the number line, so |a| ≥ 0, |−a| = |a|, and |ab| = |a||b|.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them, and raising to a power multiplies them.

Complex numbers: i² = −1; a + bi has conjugate a − bi, and (a + bi)(a − bi) = a² + b², a nonnegative real number.