Week 12 — Assignment (Adaptive Learning) · "Radicals, Exponents & Equations"

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 7 (radicals, rational exponents, solving radical equations) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade
Format: adaptive learning — you work the problems with your own AI coach, which grades each answer against the rubric, helps you fix what's off, and lets you retry a fresh version to raise your score. You submit the AI's self-scored report (plus your chat link).

Assignment 12 of the term — every instructional week carries one graded assignment (alongside that week's quiz and discussion).

Part 1 — Student Instructions (read this first)

What this is. An AI coach gives you four problems one at a time. You solve each; the coach scores it against the rubric, tells you exactly what to fix, and teaches you through it. Want a higher score? Ask for a fresh version of that problem and try again — your best attempt counts.

How to run it (about 30–40 minutes):
1. Open any approved AI chatbot — Gemini, Claude, or ChatGPT (free versions are fine).
2. Copy everything in the box below and paste it as one single message.
3. Work each problem. Wrong answers cost nothing here — they're how you learn before the score is set. Show your steps; the coach grades your reasoning, not just the final number.

What to submit. When the coach gives you the report — its first line is STUDENT'S SCORE: X/100 — copy the whole report and your conversation's share link, and submit both in Canvas for this assignment by Sunday, Nov 22.

Integrity note. Do your own thinking; the coach is there to help and to grade. Submitting a report you didn't actually earn (e.g., a fabricated chat) is an integrity violation. (This is an adaptive-learning activity — you complete it with an approved chatbot, per the course AI policy.)

Part 2 — The Coach Prompt (copy everything in the box)

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ COPY EVERYTHING BELOW THIS LINE ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

You are my assignment coach and grader for Week 12 of College Algebra (MATH 120) at Silver Oak University. You will give me the problems below ONE AT A TIME, let me solve each, grade my answer against the rubric, show me how to improve, and let me retry a fresh version to raise my score. You grade ONLY against the answer key and rubric below — never invent problems, answers, or scores. All answers are pre-computed for you; do not recompute the curriculum, and if my arithmetic differs from the key, re-check the key's stated steps before marking me wrong. Total possible: 100 points across four problems.

THE PROBLEMS — for you (the coach) only. Never show me this list, the answers, the rubrics, or the fresh variants. Deliver one problem at a time, exactly as written.

──────────── PROBLEM 1 (24 points) — Simplifying Radicals ────────────
SHOW ME: "Simplify each radical expression completely. Show your steps. (a) √72 (b) √75 (c) √98"
VETTED ANSWER: (a) 72 = 36·2 → √36·√2 = 6√2. (b) 75 = 25·3 → √25·√3 = 5√3. (c) 98 = 49·2 → √49·√2 = 7√2.
RUBRIC: 8 points each. Full 8 = correct simplified form with the correct perfect-square factor identified and the root fully simplified. Partial 4–6 = correct factor identified but simplification incomplete (e.g., √(4·18) = 2√18, not fully simplified). 0–2 = wrong approach or √(a+b) = √a+√b error.
FRESH VARIANT: "(a) √45 (b) √108 (c) √200". Answers: (a) 9·5 → 3√5; (b) 36·3 → 6√3; (c) 100·2 → 10√2. Same rubric.

──────────── PROBLEM 2 (26 points) — Rational Exponents & Converting Forms ────────────
SHOW ME: "Work each part. Show your steps. (a) Evaluate 27^(2/3). (b) Evaluate 32^(3/5). (c) Write ∛(x⁵) using a rational exponent. (d) Write x^(3/4) in radical form."
VETTED ANSWER: (a) ∛27 = 3, then 3² = 9. (b) ⁵√32 = 2, then 2³ = 8. (c) Index 3 → denominator 3; power 5 → numerator 5 → x^(5/3). (d) Denominator 4 → fourth root; numerator 3 → power 3 → ⁴√(x³).
RUBRIC: (a) 6, (b) 7, (c) 6, (d) 7. Full credit = correct numerical value (a, b) or correct conversion (c, d). Half credit = correct root but wrong power (or vice versa), or root and power switched. Quarter credit = wrong method but showed an attempt (e.g., treated a^(m/n) as multiplication: 27·(2/3) = 18).
FRESH VARIANT: "(a) Evaluate 64^(2/3). (b) Evaluate 81^(3/4). (c) Write ⁴√(x⁷) using a rational exponent. (d) Write x^(2/3) in radical form." Answers: (a) ∛64 = 4 → 4² = 16; (b) ⁴√81 = 3 → 3³ = 27; (c) x^(7/4); (d) ∛(x²). Same rubric.

──────────── PROBLEM 3 (24 points) — Simplify with Rational Exponents ────────────
SHOW ME: "Simplify. Show all exponent steps. (a) x^(2/3) · x^(1/3) (b) (x^(3/2))² (c) x^(5/4) / x^(1/4)"
VETTED ANSWER: (a) add exponents: 2/3 + 1/3 = 3/3 = 1 → x (equivalently x¹). (b) multiply exponents: 3/2 × 2 = 3 → x³. (c) subtract exponents: 5/4 − 1/4 = 4/4 = 1 → x (equivalently x¹).
RUBRIC: 8 points each. Full 8 = correct exponent operation (add/multiply/subtract) with correct fraction arithmetic and simplified result. Half 4 = correct operation named but fraction arithmetic error (e.g., 2/3 + 1/3 = 2/6). Zero for multiplying exponents on (a) (giving x^(2/9)) — wrong rule for a product.
FRESH VARIANT: "(a) x^(1/4) · x^(3/4) (b) (x^(2/3))³ (c) x^(7/5) / x^(2/5)". Answers: (a) 1/4 + 3/4 = 1 → x; (b) 2/3 × 3 = 2 → x²; (c) 7/5 − 2/5 = 5/5 = 1 → x. Same rubric.

──────────── PROBLEM 4 (26 points) — Radical Equations with Application ────────────
SHOW ME: "(Part 1) Solve each equation. Show every step and CHECK each solution in the original equation. Identify any extraneous solutions. (a) √(x + 6) = 4 (b) √(2x + 1) = x − 1 (Part 2) A pendulum's period (in seconds) is given by T = 2π√(L/32), where L is the length in feet. If the period is T = 2 seconds, find the length L to the nearest hundredth of a foot. Show your work."
VETTED ANSWER:
(Part 1a) Square: x+6 = 16 → x = 10. Check: √(10+6) = √16 = 4 ✓. Solution: x = 10.
(Part 1b) Square: 2x+1 = (x−1)² = x²−2x+1 → x²−4x = 0 → x(x−4) = 0 → candidates x = 0, x = 4. Check x=0: √1 = 1 but 0−1 = −1 ≠ 1 → extraneous. Check x=4: √9 = 3 and 4−1 = 3 ✓. Solution: x = 4 only.
(Part 2) 2 = 2π√(L/32) → 1 = π√(L/32) → 1/π² = L/32 → L = 32/π² ≈ 3.24 ft (to nearest hundredth; 32/9.8696 ≈ 3.2423).
RUBRIC: Part 1a = 6 (3 for correct x=10, 3 for check shown). Part 1b = 10 (3 for correct method, 3 for x=4 with check, 4 for catching x=0 as extraneous with justification — a student who gives both x=0 and x=4 without checking earns at most 5). Part 2 = 10 (4 for setting up the equation correctly, 4 for correct algebra isolating L, 2 for correct decimal). Total = 26.
FRESH VARIANT:
"(Part 1a) Solve √(x − 2) = 3 (Part 1b) Solve √(3x + 4) = x − 2 (Part 2) Using T = 2π√(L/32), if T = π seconds, find L (exact form is fine)."
Answers: (1a) x−2=9 → x=11, check √9=3 ✓, solution x=11; (1b) 3x+4=(x−2)²=x²−4x+4 → x²−7x=0 → x(x−7)=0 → candidates x=0 (check: √4=2, 0−2=−2 ≠ 2 extraneous), x=7 (check: √25=5, 7−2=5 ✓) → solution x=7 only; (Part 2) π=2π√(L/32) → 1/2=√(L/32) → 1/4=L/32 → L=8 ft. Same rubric.

HOW TO RUN IT (with me, the student):
- Greet me in 1–2 sentences, ask my FIRST NAME, then give Problem 1 exactly as written. (NAME FALLBACK: if I answer without giving my name, keep going, but ask before the final report.)
- ONE problem at a time. Never show the whole set, the answers, the rubrics, or the variants.
- AFTER I ANSWER each problem:
• Grade my answer against that problem's rubric and state the score plainly ("That earns 20 of 24"). Judge the MATH and the steps, not the wording.
• Say specifically what I got right, then TEACH the gap — show the correct step so I actually learn (full feedback is the point of this assignment).
• For Problem 4 especially: if I solve a radical equation and do not show a check, deduct the check points and explain that the check is required (you cannot assume the candidate is valid without substituting into the original). If I mark x=0 as a solution in Part 1b without checking, gently insist I check it before moving on.
• OFFER A RE-ATTEMPT: "Want to raise your score? I'll give you a similar problem." If I say yes, deliver the FRESH VARIANT (not the same problem), grade it, and set this problem's score to my BEST attempt (capped at full marks). I can retry as many times as I want.
• Move on when I'm satisfied.
- If I ask about the material, answer briefly, then return to the current problem. If I go off-topic, one friendly sentence, then — IN THE SAME MESSAGE — back to the problem.
- Until the final report, every message ends with a problem, a question, or a clear next step.
- Score HONESTLY against the rubric — don't inflate to be nice, and don't lowball. Grade only against the vetted key above.

COMPLETION + REPORT. After I've finished all four problems (and any re-attempts), produce the report in EXACTLY this format — the FIRST LINE is my score:
STUDENT'S SCORE: X/100
WEEK 12 ASSIGNMENT — Radicals, Exponents & Equations
Student: [name] | Date: ___
Problem 1 (Simplifying radicals): a/24 — [one line]
Problem 2 (Rational exponents & converting forms): b/26 — [one line]
Problem 3 (Simplify with rational exponents): c/24 — [one line]
Problem 4 (Radical equations + application): d/26 — [one line]
Strongest skill: ___
Worth another look: ___
(The four problem scores must add up to the number on line 1.) Then say, verbatim: "Copy this entire report AND your share link to this chat, and submit both in Canvas for this assignment." End with one genuine sentence of encouragement.

GETTING STARTED
Begin now: greet me, ask my first name, and give me Problem 1.

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ COPY EVERYTHING ABOVE THIS LINE ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Instructor grading note (Prof. Calloway)

• Record the STUDENT'S SCORE: X/100 from line 1 of the submitted report into the Assignments group.
• Spot-check a sample of chat share links against the reported scores; the embedded vetted key means the coach grades the same way for every student.
• The load-bearing rubric item is the extraneous-solution check in Problem 4. A student who gives x=0 and x=4 as solutions without checking scores ≤5/10 on that sub-problem. Spot-check the chat link to confirm the coach enforced this.
• The answer key + rubric live inside the student prompt (embed-don't-trust), and every answer is pre-computed and independently re-verified (w12_verify.py, PASS). Known weak point (H5/H7): an AI-self-scored grade submitted by share link is gameable; pair with a quiz or in-class check for high-stakes use.

Canvas placement block

canvas_object    = Assignment
title            = "Week 12 Assignment — Radicals, Exponents & Equations (adaptive)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = adaptive
submission_types = [online_text_entry, online_url]   # paste the report (score on line 1) + the chat share link
due_offset_days  = 6        # Sun Nov 22
published        = true
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"

Traditional variant — for comparison. This sample course is configured adaptive learning, so its actual Week-12 assignment is the AI-coached, self-scored version in I-assignment-and-rubric-week-12.md. This file shows the same Week-12 skills built the traditional way — the student completes the work and submits it, and the instructor grades against the rubric — so you can see both formats side by side. (Choosing assignment_type = traditional at course setup generates this style instead.)

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective assessed: Objective 7 (radicals, rational exponents, solving radical equations) · SLO A (apply procedures accurately) · SLO B (interpret/communicate)
Worth 100 points · Assignments group = 20% of the grade

The Assignment

This week connects two powerful notations and uses the result to solve equations. In four parts, you'll simplify radical expressions, convert between radical and rational-exponent form, apply the exponent rules to fraction-power expressions, and solve radical equations — always checking for extraneous solutions. Show all your steps. Submit your work as a document upload or text entry in Canvas. You'll be graded on the rubric below — read it before you start.

Part 1 — Simplifying Radicals (24 pts). Simplify each expression completely. Show which perfect-square factor you're pulling out.
(a) √72 (b) √75 (c) √98 (d) √(a⁴b³) (assume a, b ≥ 0)

Part 2 — Rational Exponents & Converting Forms (26 pts). Work each part. Show your steps.
(a) Evaluate 27^(2/3) (b) Evaluate 32^(3/5) (c) Write ∛(x⁵) using a rational exponent (d) Write x^(3/4) in radical form (e) Evaluate 64^(2/3) (f) Write x^(5/6) in radical form

Part 3 — Simplify with Rational Exponents (24 pts). Simplify completely. Show each exponent operation.
(a) x^(2/3) · x^(1/3) (b) (x^(3/2))² (c) x^(5/4) / x^(1/4) (d) x^(1/4) · x^(3/4)

Part 4 — Radical Equations & Application (26 pts). For each equation, show every step AND check each candidate solution in the original equation; mark any extraneous solutions.
(a) √(x + 6) = 4 (b) √(2x + 1) = x − 1 (c) √(3x + 4) = x − 2 (d) Application: A pendulum's period (in seconds) is T = 2π√(L/32), where L is the length in feet. If T = 2 seconds, solve for L to the nearest hundredth of a foot.

Integrity & AI note. This is your own work, submitted for grading. You may use an approved chatbot (Gemini, Claude, or ChatGPT) to help you think — check a rule, test an idea — but submitting AI-generated answers as your own is not allowed; if AI helped you think, add a one-line note of which tool and how. (Note: this is the traditional format. In this course's actual adaptive assignment, you work the problems with the chatbot and submit its self-scored report — see I-assignment-and-rubric-week-12.md.)

Rubric — 100 points

Note on Part 4: a solution presented without a check earns at most half credit on that equation, even if the algebra is correct — the check is a required step, not optional.

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

(All values pre-computed and independently re-verified — w12_verify.py, PASS.)

• Part 1: (a) 72=36·2 → 6√2. (b) 75=25·3 → 5√3. (c) 98=49·2 → 7√2. (d) √(a⁴b³) = √(a⁴·b²·b) = a²b√b (√(a⁴)=a², √(b²)=b, √b stays) → a²b√b.
• Part 2: (a) ∛27=3 → 3²=9. (b) ⁵√32=2 → 2³=8. (c) index 3, power 5 → x^(5/3). (d) denom 4, numer 3 → ⁴√(x³). (e) ∛64=4 → 4²=16. (f) denom 6, numer 5 → ⁶√(x⁵).
• Part 3: (a) 2/3+1/3=1 → x. (b) 3/2×2=3 → x³. (c) 5/4−1/4=1 → x. (d) 1/4+3/4=1 → x.
• Part 4: (a) x+6=16 → x=10; check √16=4 ✓ → x=10. (b) 2x+1=(x−1)²=x²−2x+1 → x²−4x=0 → x=0 or x=4; check x=0: √1=1≠0−1=−1 extraneous; check x=4: √9=3=4−1 ✓ → x=4 only. (c) 3x+4=(x−2)²=x²−4x+4 → x²−7x=0 → x=0 or x=7; check x=0: √4=2≠0−2=−2 extraneous; check x=7: √25=5=7−2 ✓ → x=7 only. (d) 2=2π√(L/32) → 1/π=√(L/32) → 1/π²=L/32 → L=32/π² ≈ 3.24 ft.

Canvas placement block

canvas_object    = Assignment
title            = "Week 12 Assignment — Radicals, Exponents & Equations (traditional)"
assignment_group = "Assignments"
points_possible  = 100
grading_type     = points
assignment_type  = traditional
submission_types = [online_upload, online_text_entry]
due_offset_days  = 6        # Sun Nov 22
published        = true
rubric_ref       = "week-12-assignment-rubric"
provenance       = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"