Week 12 — Roots, Fraction Exponents, and the Answer That Fails: A Short Radical Report — Traditional alternative (no AI)

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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 week you learned to read a root as the undo of a power, to simplify a square root with the product rule, to use the quotient rule, to read a rational exponent as a root taken first, and to solve a radical equation while checking every candidate. 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. In this assignment you work four short problems, interpret each one, and explain to someone who has never studied radicals why a squared equation can give an answer that does not work. This week's lecture tutorial, quiz, discussion, and Model/Build Studio practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single radical report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of what that value shows and one use or limit of it;
• your message for your family member, explaining how squaring creates the false answer, how substitution catches it, and one honest caveat; and
• a one-line restatement of the expressions and equations you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch with a short note. Every form is graded by the same scoring criteria. 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.

How you'll be graded

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

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a free graphing calculator such as Desmos to check your arithmetic and your answers. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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