Use the Sign of an Odd Power

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Find the real cube root of −8. Candidate answers: (A) -2 (B) -4 (C) -8 (D) 2.
• Independent check inputs: Find the real cube root of −27. Candidate answers: (A) -27 (B) -3 (C) -9 (D) 3.

Learn the idea

Roots reverse powers, but their exact values and decimal approximations must be distinguished. Integer squares provide reliable bounds, and odd roots retain negative signs while principal square roots are nonnegative. A real cube root is the unique real number whose cube is the radicand. A negative factor multiplied three times stays negative, so negative radicands have negative real cube roots.

Here is the method on the supplied teaching case. Because (−2)³=−8, the cube root is −2. Odd powers preserve the sign, unlike even powers.

Do not apply the nonnegative principal-square-root convention to cube roots. Verify by cubing the answer; this gives an exact check and distinguishes the root from an arbitrary division by three.

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: Find the real cube root of −729. Candidate answers: (A) -729 (B) -81 (C) -9 (D) 9. Show the mathematical reason, not only the selected answer. Expected result: -9. Because (−9)³=−729, the cube root is −9. Odd powers preserve the sign, unlike even powers.

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