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BetterGrades Algebra · Unit A11 · Unit Hub

Unit A11: Radicals, Rational Exponents, and Complex Numbers

Develop exact root arithmetic, radical equations, extraneous-solution logic, and basic complex arithmetic. How do we reverse powers, and what happens when the real numbers are insufficient?

Act XI - Reversing powers and enlarging the number system

The 10-lesson path

General roots and rational exponents express inverse powers. Radical arithmetic creates exact forms, conjugates rationalize denominators, and raising powers solves equations but may introduce candidates. Complex numbers complete quadratic solutions.

Governing question

How do we reverse powers, and what happens when the real numbers are insufficient?

Develop exact root arithmetic, radical equations, extraneous-solution logic, and basic complex arithmetic.

Mastery design

Know when to continue

Students preserve domains, simplify exact forms, check radical candidates, and express nonreal quadratic roots correctly.

Unit outcomes
  1. Interpret principal and general roots.
  2. Simplify and combine radical expressions.
  3. Rationalize denominators using factors or conjugates.
  4. Connect radicals and rational exponents.
  5. Solve radical equations and perform basic complex arithmetic.
Core sequence

Lessons

Move in order when this material is new; use prerequisites and repair links when entering mid-unit.

  1. 01General nth roots and principal rootsDistinguish the principal root operation from finding every solution of a power equation.
  2. 02Simplifying radicalsExtract perfect-power factors and retain exact form.
  3. 03Products and quotients of radicalsApply root product and quotient properties under valid real-number conditions.
  4. 04Adding and subtracting radicalsSimplify first and combine only like radical terms.
  5. 05Rationalizing monomial denominatorsMultiply by a form of one to remove a radical denominator.
  6. 06Conjugates and binomial denominatorsUse conjugates and difference of squares to remove radical binomial denominators.
  7. 07Rational exponentsInterpret the denominator of an exponent as a root and the numerator as a power.
  8. 08Solving radical equationsIsolate radicals and raise powers while treating results as candidates.
  9. 09Extraneous solutions and domainExplain why nonreversible operations can enlarge a solution set.
  10. 10Complex numbers and conjugatesExtend arithmetic consistently using i2=−1i^2=-1 and represent nonreal roots.

Practice around the path

Review, mixed practice, mastery, and investigation

Every route contains a fixed set of concrete questions, with detailed solutions available after an attempt.

ReviewCumulative reviewRevisit this unit with prior-unit retrieval.PracticeMixed practiceClassify the structure before choosing a method.MasteryMastery checkUse the supplied grading and cumulative-share contract.Response guideMastery response guideAttempt-gated rubric support; no invented canonical answers.

Source & rights

Original storyboard, rights-separated references.

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