Extract a Perfect-Square Factor

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Simplify √48 exactly. Candidate answers: (A) 24√3 (B) 2√3 (C) 4√3 (D) 4√6.
• Independent check inputs: Simplify √108 exactly. Candidate answers: (A) 36√3 (B) 3√3 (C) 6√3 (D) 6√6.

Learn the idea

Root expressions can be simplified through perfect-power factors and rational exponents. Squaring equations can introduce extraneous candidates, while complex arithmetic extends the number system using an imaginary unit with square negative one. A nonnegative radical can be simplified by separating a perfect-square factor. The square root of that factor comes outside, leaving a radicand with no square factor remaining.

Here is the method on the supplied teaching case. 48=(4)²·3. Taking the positive root of the perfect-square factor gives 4√3, whose remaining radicand has no square factor.

Keep the factor outside positive for these principal square roots. Do not split sums inside radicals; the product rule used here requires nonnegative factors and is different from √(u+v).

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Simplify √972 exactly. Candidate answers: (A) 108√3 (B) 18√3 (C) 18√6 (D) 9√3. Show the mathematical reason, not only the selected answer. Expected result: 18√3. 972=(18)²·3. Taking the positive root of the perfect-square factor gives 18√3, whose remaining radicand has no square factor.

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