Week 12 — Roots, Fraction Exponents, and the Answer That Fails: A Short Radical Report

Assignment 11 of the term — simplify a root, evaluate a fraction exponent, time a falling object, catch an extraneous candidate, and tell someone new to algebra why a squared equation can hand you an answer that never worked.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, checkable report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. In Week 1 you learned the exponent rules and the powers of two, and in Week 11 you met extraneous solutions in rational equations; this week both return with roots. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles; this assignment is where you put them together and explain them.

The task

You will use four pieces of this week's work: the square root of 48, the fraction exponent 16^(3⁄4), the dropped-object formula t = √(h ⁄ 16) for an object dropped from rest that falls h feet in t seconds, and the radical equation √(x + 3) = x + 1.

Work these four problems, showing your arithmetic:

1. Simplify √48 with the product rule. What is the largest perfect-square factor of 48 that you pull out from under the root?
2. Evaluate 16^(3⁄4), taking the root first. What is its value?
3. An object dropped from rest falls h feet in t = √(h ⁄ 16) seconds. How many seconds does a drop from 144 feet take?
4. Squaring both sides of √(x + 3) = x + 1 gives two candidates. What is the candidate that fails the check, the extraneous solution?

Beside each answer, add one brief interpretation: what that value shows and one use or limit of it. For problem 1, write √48 in simplified radical form, check it by squaring, and say why 2√12 is not finished. For problem 2, name which number in the exponent is the root and which is the power, check the value with the power rule using 16 = 2⁴, and say why 16 × 3⁄4 = 12 is wrong. For problem 3, show the quotient rule step and say what the model ignores. For problem 4, name both candidates, check each one in the original equation, and say why the failing one fails.

Explain it to a non-expert

A family member is helping you check homework and has never studied radicals. They watched you square √(x + 3) = x + 1, get two numbers, and then cross one out, and they want to know which answer to trust. Write a 3–5 sentence message: why squaring both sides can create an answer that does not work, how substituting into the original equation catches it, and one honest limit or caveat, such as why the check must use the original equation and not the squared one. Explain any technical term in everyday words. More than one good caveat can be defended.

For a different, ungraded example, imagine the equation √x = −3. Model answer: "A square root sign always means the positive root, so it can never come out negative. If you square both sides anyway, you get x = 9, which looks like an answer. But the square root of 9 is 3, not negative 3, so 9 does not really work. Squaring hides the minus sign, which is why every answer has to go back into the original equation."

Use that pattern for √(x + 3) = x + 1, with your own explanation.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message for your family member before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about radicals and rational exponents: the index and radicand, the product and quotient rules for radicals, simplified radical form, why a root never splits over a sum, rational exponents taken root first, radical equations, and extraneous solutions. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: the square root of 48, the fraction exponent 16^(3/4), the dropped-object formula t = sqrt(h / 16) for an object dropped from rest that falls h feet in t seconds, and the radical equation sqrt(x + 3) = x + 1.
PROBLEM P1: Simplify √48 with the product rule. What is the largest perfect-square factor of 48 that you pull out from under the root?
PROBLEM P2: Evaluate 16^(3⁄4), taking the root first. What is its value?
PROBLEM P3: An object dropped from rest falls h feet in t = √(h ⁄ 16) seconds. How many seconds does a drop from 144 feet take?
PROBLEM P4: Squaring both sides of √(x + 3) = x + 1 gives two candidates. What is the candidate that fails the check, the extraneous solution?
After the four answers, I must briefly interpret each one: what that value shows and one use or limit of it; for P1, sqrt(48) in simplified radical form, checked by squaring, and why 2 sqrt(12) is not finished; for P2, which number in the exponent is the root and which is the power, a check with the power rule using 16 = 2^4, and why 16 * 3/4 = 12 is wrong; for P3, the quotient rule step and what the model ignores; for P4, both candidates, each checked in the original equation, and why the failing one fails. I must then write a 3-5 sentence message for a family member who is new to radicals, explaining why squaring both sides can create an answer that does not work, how substituting into the original equation catches it, and one honest limit or caveat. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct radical and exponent computations (40 points), reasoning about roots, exponents, and candidates (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct perfect-square factor, exponent value, fall time, or failing candidate = 2; the right values and operation with only an arithmetic slip = 1; a missing result or the wrong values or operation = 0. Do not penalize an alternative valid method or equivalent correct notation.
REASONING UNITS: For P1-P3 separately, an accurate account of what that value shows plus a use or limit = 2; exactly one of the two = 1; neither = 0. For P1, a perfect-square factor without the simplified form does not count as the account; for P2, a value without naming the root and the power does not count as the account. For P4, BOTH an accurate check of both candidates with the reason the failing one fails AND a final message that explains how squaring creates the false answer, how substitution catches it, and an honest caveat = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to algebra, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message for my family member AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised radical report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised radical report with all four answers, their brief interpretations, and your message for your family member — written, or recorded as audio or video with a transcript, or as a labelled sketch with a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

Equivalent route (no AI required)

This route assesses the same Week 12 skills — simplifying a square root with the product rule, evaluating a rational exponent root first and checking it with the exponent rules, using the quotient rule in a setting, and solving a radical equation while rejecting an extraneous solution — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message for your family member. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised radical report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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