Free Praxis Mathematics Exam Flashcards

A rational exponent encodes both a root and a power: a^(m/n) equals the n-th root of a raised to the m power. Computing the root first, then raising to the exponent, avoids unwieldy intermediate values when possible.

The real numbers satisfy closure, commutativity, associativity, distributivity, identity, and inverse properties for both addition and multiplication. These axioms justify every algebraic manipulation used to solve equations and simplify expressions.

When multiplying complex numbers, distribute as with binomials and replace i-squared with -1. For (a + bi)(c + di), the product expands to (ac - bd) + (ad + bc)i after combining like terms.

When k is non-negative, the equation |A| = k splits into two cases: A = k and A = -k. If k is negative, there is no solution because absolute value is always non-negative.

Multiply by ratios equal to 1 (such as 1 meter over 100 cm) so unwanted units cancel. Arrange the chain so the starting unit appears once in a numerator and once in a denominator, leaving only the desired target unit.

A ratio compares two quantities of the same unit, a rate compares quantities of different units, and a proportion is an equation stating two ratios are equal. Solving a proportion uses cross-multiplication to isolate the unknown.

Every positive real number has one real n-th root when n is odd, and two real n-th roots (positive and negative) when n is even. The radical symbol denotes the principal (non-negative) root by convention.

The expression a-squared minus b-squared factors as (a - b)(a + b). This identity is reversible: recognizing the pattern in expanded form signals a difference of squares that can be factored immediately.

For ax-squared + bx + c = 0 with a not zero, the solutions are x = (-b plus or minus the square root of b-squared minus 4ac) over 2a. The discriminant (b-squared minus 4ac) determines whether roots are real and distinct (positive), repeated (zero), or complex (negative).

Rewrite ax-squared + bx + c by factoring out a, then adding and subtracting (b over 2a)-squared inside the parentheses. This produces the vertex form a(x - h)-squared + k, which reveals the vertex (h, k) directly.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol. Adding, subtracting, or multiplying by a positive number preserves the original direction of the inequality.

Values that make the denominator zero are excluded from the domain even after simplification. The original restrictions carry forward: for example, (x-squared - 9) over (x-squared - 3x) simplifies to (x + 3) over x but still forbids x = 0 and x = 3.

Every non-constant polynomial with complex coefficients has at least one complex root. A degree-n polynomial therefore has exactly n roots when counting multiplicity over the complex numbers.

Elimination adds or subtracts equations to cancel a variable; it is strongest when coefficients align. Substitution isolates one variable and plugs into the other equation; it is strongest when one equation is already solved for a variable.

A number r is a zero of polynomial p(x) when p(r) = 0. By the Factor Theorem, this is equivalent to (x - r) being a factor of p(x), which lets you construct a polynomial from its known roots or find roots by testing factors.

A function assigns each element of the domain to exactly one element of the range. If any input maps to two or more outputs, the relation is not a function; the vertical-line test detects this on a graph.

The domain is the set of all valid inputs; the range is the set of all resulting outputs. For real-valued functions, domain excludes values causing division by zero or even roots of negatives, while range depends on the function family.

Composition applies g first, then f, so (f composed with g)(x) = f(g(x)). The domain of the composition is restricted to inputs x where g(x) is defined and g(x) lies in the domain of f, which can be narrower than either function alone.

An inverse function reverses the mapping of f, so that the inverse of f(x) equals x. A function has an inverse only if it is one-to-one (passes the horizontal-line test); otherwise the domain must be restricted to create a valid inverse.

An even function satisfies f(-x) = f(x), producing y-axis symmetry. An odd function satisfies f(-x) = -f(x), producing origin symmetry. Most functions are neither, but the distinction simplifies graphing and integration on symmetric intervals.

In f(x) + k the shift is vertical (up for positive k), while in f(x - h) the shift is horizontal (right for positive h). The horizontal direction is counterintuitive because subtracting h inside the function moves the graph right, not left.

An exponential function has a constant base b greater than 0 and not equal to 1, with a variable exponent. When b exceeds 1 the function grows; when b is between 0 and 1 it decays. The y-intercept is always a. Common applications include compound interest, population growth, and radioactive decay.

The logarithm log-base-b of y equals x means b to the x power equals y, so logarithms undo exponentials. Key identities: log of a product equals sum of logs, log of a quotient equals difference of logs, and log of x to the n equals n times log of x.

On the unit circle, cosine gives the x-coordinate and sine gives the y-coordinate of the point at angle theta. This generalizes sine and cosine beyond right triangles to any real angle, including negative angles and values beyond 90 degrees.

Over the interval from a to b, the average rate of change of f is (f(b) - f(a)) divided by (b - a). Geometrically, this is the slope of the secant line connecting two points on the graph, and by the Mean Value Theorem it equals the instantaneous rate at some point in between.

The derivative f-prime of x is the limit of the difference quotient [f(x+h) - f(x)] over h as h approaches 0, when that limit exists. It gives the slope of the tangent line at x and represents the instantaneous rate of change of the function.

For any real number n, the derivative of x to the n is n times x to the (n minus 1). This rule extends to negative and fractional exponents, so the derivative of x to the one-half is one-half times x to the negative one-half, and the derivative of x to the negative 3 is negative 3 times x to the negative 4.

The definite integral from a to b of f(x) dx represents the net signed area between the curve and the x-axis: regions above the axis count positive and regions below count negative. If f is non-negative on the interval, the integral equals the ordinary geometric area.

This theorem links differentiation and integration in two parts. First, if F is an antiderivative of f, then the integral from a to b of f(x) dx equals F(b) minus F(a). Second, the derivative of the integral from a to x of f(t) dt equals f(x), showing differentiation undoes integration.

A critical point occurs where the derivative equals zero or does not exist. Local maxima and minima can only occur at critical points or endpoints of the domain, and the first or second derivative test determines which type each critical point yields.

Two triangles are similar if two pairs of corresponding angles are congruent, because the third angles are then automatically equal. Similar triangles have proportional corresponding side lengths but not necessarily equal side lengths.

In a right triangle with legs a and b and hypotenuse c, the relationship a-squared plus b-squared equals c-squared holds. The converse is also true: if a-squared plus b-squared equals c-squared, then the triangle is a right triangle with the right angle opposite side c.

An inscribed angle in a circle measures half of the central angle that subtends the same arc. A key special case: an angle inscribed in a semicircle is always a right angle because the corresponding central angle is 180 degrees.

The distance between points (x1, y1) and (x2, y2) is the square root of (x2 - x1)-squared plus (y2 - y1)-squared. This is the Pythagorean theorem applied to the horizontal and vertical legs of the right triangle formed by the two points.

SSS, SAS, and ASA each guarantee two triangles are congruent; AAS also works because the third angle is forced by the angle sum. SSA and AAA do not guarantee congruence, which is why they are excluded from the standard criteria.

When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary. These angle relationships are foundational for proving lines parallel and for coordinate geometry slope arguments.

A circle with radius r has circumference C = 2 times pi times r and area A = pi times r-squared. Circumference is the one-dimensional boundary measure while area is the two-dimensional enclosed space; each formula depends only on the radius.

A cylinder of radius r and height h has volume pi times r-squared times h. A cone with the same base and height has one-third of that volume. A sphere of radius r has volume four-thirds times pi times r-cubed. The cone-to-cylinder ratio of 1 to 3 is a commonly tested relationship.

Translations, rotations, and reflections are rigid transformations that preserve distances and angles. Dilations are non-rigid, preserving angles but scaling distances. Congruence is preserved by rigid transformations alone, while similarity is preserved by both rigid and non-rigid transformations.

Slope-intercept form y = mx + b reveals slope m and y-intercept b directly. Point-slope form y minus y1 = m times (x minus x1) is preferable when you know a point on the line and the slope but not the y-intercept.

The mean is the arithmetic average, the median is the middle value when data is ordered, and the mode is the most frequent value. The mean is sensitive to outliers, the median is resistant to outliers, and the mode applies to categorical data as well as numerical data.

Variance is the average of squared deviations from the mean; standard deviation is the square root of variance, restoring the original units. A larger standard deviation signals greater spread in the data, and it is always non-negative.

A normal distribution is symmetric and bell-shaped, with mean equal to median equal to mode. The empirical rule states that about 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three.

Correlation measures the strength and direction of an association between two variables, but it does not establish that one variable causes the other. A lurking variable or coincidence can produce strong correlation with no causal link, so controlled experiments are needed to establish causation.

The sample space is the set of all possible outcomes of a random experiment. For equally likely outcomes, the probability of an event E is the number of favorable outcomes divided by the total number of outcomes, always yielding a value between 0 and 1 inclusive.

Conditional probability is P(A given B) = P(A and B) divided by P(B), the probability of A given that B has occurred. It restricts the sample space to B and renormalizes accordingly, which can increase or decrease the probability of A relative to its unconditional value.

Two events are independent when the occurrence of one does not change the probability of the other: P(A given B) equals P(A). For independent events, P(A and B) equals P(A) times P(B); for dependent events, the intersection probability must account for the conditioning.

The expected value of a random variable is the long-run average outcome, computed as the sum of each value times its probability over all possible values. It need not equal any single possible outcome, as with a fair six-sided die whose expected value is 3.5.

Permutations count ordered arrangements: nPr = n factorial over (n minus r) factorial. Combinations count unordered selections: nCr = n factorial over r factorial times (n minus r) factorial. Use permutations when order matters (rankings, codes) and combinations when it does not (committees, subsets).

A simple random sample gives every member of the population an equal chance of selection, supporting valid statistical inference. Common biases include voluntary response, convenience sampling, and undercoverage, all of which can distort conclusions even with large sample sizes.