Find the rule.
Know when to use it.

Multiply the outside factor by every term inside the grouping symbols.

Expanding expressions or factoring out a common factor.

Changing the order of added terms does not change the sum.

Reordering terms to calculate or simplify more easily.

Changing the order of factors does not change the product.

Reordering factors to simplify a product.

Regrouping addition or multiplication does not change the result.

Combining convenient groups without changing their order.

Use a common denominator before adding numerators.

Adding or subtracting fractions with unlike denominators.

Multiply by the reciprocal of the divisor.

Any division involving fractions, provided the divisor is nonzero.

Compare the amount of change with the original amount.

Growth, decrease, markup, discount, or relative change.

Equal ratios have equal cross-products when both denominators are nonzero.

Solving scale, rate, similarity, and conversion problems.

m is the slope and b is the y-intercept.

Graphing a line or reading its rate and starting value.

A line is determined by a slope and one point.

Writing a line when a point and slope are known.

For ax² + bx + c = 0, this returns every real or complex solution.

Solving a quadratic that is difficult or impossible to factor.

Positive gives two real roots, zero gives one repeated real root, and negative gives two complex roots.

Predicting the number and type of quadratic solutions.

A subtraction of two perfect squares factors into conjugates.

Fast factoring and simplifying rational expressions.

The middle term is twice the product of the square roots of the outer terms.

Factoring or completing the square.

When bases match, add exponents while multiplying.

Simplifying products with the same base.

When nonzero bases match, subtract denominator exponents from numerator exponents.

Simplifying quotients with the same base.

Multiply exponents when raising a power to another power.

Simplifying nested exponents.

A negative exponent indicates the reciprocal of the positive power.

Rewriting expressions without negative exponents.

A logarithm turns multiplication into addition.

Expanding logarithms or solving exponential equations.

An exponent inside a logarithm becomes a factor outside.

Expanding, condensing, or solving logarithmic equations.

Each term changes by the same common difference d.

Finding a term in a linear sequence.

Each term is multiplied by the same common ratio r.

Finding a term in an exponential sequence.

This sums the first n terms when r is not 1.

Adding a finite geometric pattern without listing every term.

Slope is vertical change divided by horizontal change.

Finding a line’s rate of change from two points.

Coordinate distance is the Pythagorean theorem applied to horizontal and vertical changes.

Finding the straight-line distance between two points.

Average corresponding coordinates.

Finding the center of a segment.

The center is (h, k) and the radius is r.

Graphing or analyzing a circle on a coordinate plane.

In a right triangle, the hypotenuse square equals the sum of the leg squares.

Finding a missing side of a right triangle.

Use a base and its perpendicular height.

Finding the area of any triangle.

Circumference measures the boundary; area measures the enclosed region.

Circle length and region problems.

An arc is the same fraction of the circumference as its central angle is of 360°.

Finding a portion of a circle’s circumference.

A sector is the same fraction of the circle’s area as its central angle is of 360°.

Finding the area of a pie-shaped circle region.

An n-sided polygon can be divided into n − 2 triangles.

Finding the total interior-angle measure of a polygon.

Multiply the area of a constant cross-section B by the perpendicular height.

Volumes of prisms and cylinders.

A pyramid or cone has one third the volume of a matching prism or cylinder.

Volumes of pyramids and cones.

Both measures depend only on the sphere’s radius.

Surface-area or volume problems involving spheres.

Sine, cosine, and tangent compare specific side pairs relative to an angle.

Finding missing sides or angles in right triangles.

Sine and cosine coordinates lie on the unit circle.

Rewriting and simplifying trigonometric expressions.

Each side-to-opposite-angle ratio is equal.

Non-right triangles with an opposite side-angle pair.

This extends the Pythagorean theorem to any triangle.

Non-right triangles with SAS or SSS information.

Multiply degrees by π/180 to get radians; multiply radians by 180/π to get degrees.

Converting angle units.

An event and its complement cover every possible outcome without overlap.

“At least one,” “not,” and easier opposite-event problems.

Subtract the overlap because it was counted twice.

The probability that A or B occurs.

For independent events, one event does not change the probability of the other.

The probability that independent events both occur.

Restrict the sample space to outcomes where B occurred.

Problems containing “given that” or dependent information.

Reverse a conditional probability by combining likelihood and prior probability.

Updating the probability of a cause after observing evidence.

Order matters when selecting and arranging r objects from n.

Rankings, arrangements, or ordered selections.

Order does not matter when choosing r objects from n.

Teams, groups, hands, and unordered selections.

Counts exactly k successes in n independent trials with constant success probability p.

Repeated yes/no trials with a fixed number of attempts.

Weight every possible value by its probability.

Long-run averages for a discrete random variable.

Add every observation and divide by the number of observations.

Describing the center of data without strong outliers.

Values with larger weights contribute more to the mean.

Grades, mixtures, and data with unequal importance or frequency.

Average squared distance from the sample mean using n − 1 degrees of freedom.

Estimating population spread from a sample.

Measure how many standard deviations a value lies from the mean.

Comparing values from a normal distribution or different scales.

Sampling uncertainty decreases as sample size grows.

Confidence intervals and inference about a population mean.

Estimate ± critical value × standard error.

Estimating a population mean when population standard deviation is unknown.

Bring the exponent down as a coefficient, then subtract one from the exponent.

Differentiating polynomial powers.

Differentiate one factor at a time while keeping the other.

Differentiating a product of two functions.

Low times derivative of high minus high times derivative of low, over low squared.

Differentiating one function divided by another.

Differentiate the outside function, then multiply by the derivative of the inside.

Differentiating nested or composed functions.

Add one to the exponent and divide by the new exponent, when n ≠ −1.

Integrating polynomial powers.

Evaluate an antiderivative F at the upper and lower limits.

Computing an exact definite integral.

Divide total accumulated value by interval length.

Finding a continuous function’s mean value on an interval.