Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
Page 19 of 22
Each problem type has four distinct practice variants. Open a preview to move among all four.
Simplify a power raised to a power when the exponents are rational numbers
Rewrite expressions with radicals and rational exponents using exponent properties.
A power raised to another power multiplies its exponents rather than adding them. We’ll express the outside integer as a fraction, multiply and cancel common factors, state the resulting whole-number …
Preview problemRewrite a product of radicals with the same base by adding rational exponents
Rewrite expressions with radicals and rational exponents using exponent properties.
Radicals with the same radicand become like-base powers after conversion to rational exponents. We’ll translate each root index, add the exponents with a common denominator, reduce the sum, and preserve …
Preview problemRewrite a rational exponent in radical form
Rewrite expressions with radicals and rational exponents using exponent properties.
Rational-exponent conversion assigns the denominator to the radical index and numerator to the radicand’s power. We’ll map those roles without swapping them, construct the radical, convert back through a power …
Preview problemSimplify a radical by factoring out a perfect square or perfect cube
Rewrite expressions with radicals and rational exponents using exponent properties.
Simplifying a radical starts with the greatest perfect-power factor that matches its root index. We’ll factor the radicand, split the root across the product, evaluate the extractable square, leave the …
Preview problemSimplify radicals with variable exponents by pulling out perfect powers
Rewrite expressions with radicals and rational exponents using exponent properties.
Variable powers leave the radical in complete pairs determined by the root index. We’ll separate the greatest even exponent, establish the real domain, take the principal root of the paired …
Preview problemRationalize the denominator when the denominator has a single square root factor
Rewrite expressions with radicals and rational exponents using exponent properties.
Rationalizing a single radical denominator means multiplying by a carefully chosen form of one. We’ll use the same radical in numerator and denominator, confirm the value is preserved, simplify the …
Preview problemDecide whether a radical expression and a rational-exponent expression are equivalent
Rewrite expressions with radicals and rational exponents using exponent properties.
Equivalence of radical and exponent forms must be judged on their shared real domain. We’ll determine where the whole radicand is nonnegative, rewrite the square root as a one-half power, …
Preview problemSolve equations of the form x^(m/n) = k in real numbers
Rewrite expressions with radicals and rational exponents using exponent properties.
The one-half power is a principal square root, so its output and real radicand are nonnegative. We’ll rewrite the equation in radical form, square both sides under those valid conditions, …
Preview problemChoose a form by an explicit direct-operation criterion
Rewrite expressions with radicals and rational exponents using exponent properties.
The most useful form is the one that exposes an exact first operation. We’ll read the denominator as a root index and numerator as a power, rewrite the expression so …
Preview problemClassify the sum of two rational numbers as rational
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Rational numbers are closed under addition, so classification can be established before arithmetic. We’ll identify both addends as integer ratios with nonzero denominators, invoke closure, compute a common-denominator sum, and …
Preview problemClassify the product of two rational numbers as rational or irrational
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
The product of two rational numbers remains rational by closure, whether or not it is an integer. We’ll classify each factor from its integer-ratio form, multiply numerators and denominators, reduce …
Preview problemClassify a sum of a rational number and an irrational number
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Adding a rational number cannot turn an irrational number rational. We’ll classify each addend, suppose the sum were rational, subtract the rational addend using closure, and expose the contradiction that …
Preview problemClassify a rational-irrational product with an explicit zero check
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
The rational-times-irrational rule needs an explicit nonzero condition. We’ll classify both factors, check that the rational factor is not zero, apply the qualified rule, and verify it by showing that …
Preview problemAudit a zero exception in a quantified product statement
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
A universal claim can be disproved by one valid counterexample, so the zero case must be tested. We’ll confirm zero is rational, multiply it by an arbitrary irrational number, classify …
Preview problemSimplify and classify a concrete sum of irrational expressions
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Two irrational terms can have a rational sum when their irrational parts cancel. We’ll recognize the additive-inverse structure, combine the radical coefficients, simplify the result exactly, classify that value as …
Preview problemSimplify and classify a concrete product of irrational expressions
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Multiplying a principal square root by itself returns its radicand, so irrational factors need not produce an irrational product. We’ll apply that identity, verify it through the radical product property, …
Preview problemClassify a radical expression as rational or irrational after simplifying
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
Classification should follow exact simplification, not the appearance of separate radicals. We’ll check the quotient-property conditions, combine the quotient under one square root, simplify the radicand, evaluate the principal root, …
Preview problemClassify always, sometimes, or never statements about rational and irrational sums and products
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.
An always statement needs a proof for arbitrary inputs, not a handful of examples. We’ll represent two rational numbers as integer ratios, add them with a common denominator, prove the …
Preview problemList the sample space of a finite experiment
Describe events as subsets of a sample space using unions, intersections, and complements.
A compound outcome must record one result from each stage of the experiment. We’ll list the component possibilities, pair every coin result with every spinner result, remove no valid combinations, …
Preview problemList the outcomes that belong to an event in a sample space
Describe events as subsets of a sample space using unions, intersections, and complements.
An event is the subset of sample-space outcomes satisfying its defining condition. We’ll retain the complete sample space, translate even into divisibility by two, test every listed outcome, and write …
Preview problemFind the union of two events written as sets of outcomes
Describe events as subsets of a sample space using unions, intersections, and complements.
Union uses inclusive-or membership: an outcome belongs if it appears in either event or both. We’ll begin with every member of the first set, add the second set’s new members, …
Preview problemFind the intersection of two events
Describe events as subsets of a sample space using unions, intersections, and complements.
Intersection requires simultaneous membership in both events. We’ll test each outcome from one set against the other, discard members appearing in only one place, confirm there are no additional shared …
Preview problemFind the complement of an event in a sample space
Describe events as subsets of a sample space using unions, intersections, and complements.
A complement is always defined relative to the supplied sample space. We’ll compare the event against that universe, remove every event member, list the remaining outcomes, and check both defining …
Preview problemSort a sample space into the four regions of a two-event Venn diagram
Describe events as subsets of a sample space using unions, intersections, and complements.
The four Venn regions form a disjoint partition of the sample space. We’ll find the overlap first, remove it to obtain each event-only region, subtract the union from the sample …
Preview problemInterpret union, intersection, and complement notation in words
Describe events as subsets of a sample space using unions, intersections, and complements.
Union translates to inclusive or, so the overlap remains part of the event. We’ll identify the operation, state its either-or-both membership rule, translate it into words, and contrast it with …
Preview problemWrite a verbal event description using union, intersection, or complement notation
Describe events as subsets of a sample space using unions, intersections, and complements.
The phrase or in event language is inclusive unless the problem explicitly excludes overlap. We’ll translate the membership condition, match it to the operation that includes either event and both, …
Preview problemCount the number of outcomes in a union or intersection event
Describe events as subsets of a sample space using unions, intersections, and complements.
Counting a union means counting distinct outcomes, not adding both event sizes blindly. We’ll build the union without duplicates, count its members, identify the shared outcomes, and verify the total …
Preview problemDecide whether two events are mutually exclusive by checking their overlap
Describe events as subsets of a sample space using unions, intersections, and complements.
Mutual exclusivity is a statement about overlap, not about whether events share a sample space. We’ll list each die event, test for any common outcome, express the intersection, and apply …
Preview problemDecide whether two events are exhaustive by checking their union against the sample space
Describe events as subsets of a sample space using unions, intersections, and complements.
Exhaustive events cover every possible outcome, whether or not they overlap. We’ll list the die’s sample space, unite the two events, compare that union member by member with the universe, …
Preview problemDetermine whether two events are independent using \(P(A and B) = P(A)P(B)\)
Determine event independence using P(A and B)=P(A)P(B).
Independence is tested by comparing the joint probability with the product of the two marginal probabilities. We’ll multiply the given fractions, simplify the product, place it beside the observed intersection …
Preview problemFind P(A and B) when events are independent
Determine event independence using P(A and B)=P(A)P(B).
For independent events, the multiplication rule directly supplies their intersection probability. We’ll substitute the two marginals, multiply numerators and denominators, reduce the fraction, and check that the joint result does …
Preview problemDecide whether two events are independent by comparing P(A and B) with P(A)P(B)
Determine event independence using P(A and B)=P(A)P(B).
Counts must first be normalized by the common total before independence is tested. We’ll convert the two event counts and overlap count to probabilities, multiply the marginals, reduce both sides …
Preview problemDetermine whether two events are independent by checking whether P(A and B) = P(A)P(B)
Determine event independence using P(A and B)=P(A)P(B).
Venn marginals include each event’s exclusive region plus the overlap. We’ll compute both full event probabilities and the observed joint probability, multiply the marginals, and compare the two exact joint …
Preview problemTest two events for independence using joint and marginal probabilities
Determine event independence using P(A and B)=P(A)P(B).
Separate random devices suggest independence, but the probability equation provides the proof. We’ll define the coin and die events, record their marginals and joint probability, multiply the marginals, and interpret …
Preview problemClassify independence and mutual exclusivity with separate probability tests
Determine event independence using P(A and B)=P(A)P(B).
Mutual exclusivity and independence require separate tests. We’ll use zero overlap to classify exclusivity, multiply the positive marginal probabilities for the independence benchmark, compare that nonzero product with the zero …
Preview problemFind the probability of the same independent event happening repeatedly
Determine event independence using P(A and B)=P(A)P(B).
All required outcomes across independent trials form an intersection, so their probabilities multiply. We’ll express one success for each flip, write the repeated product as a power, evaluate it, and …
Preview problem