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Grade 11 · lesson 2 of 4

Interpret a Fractional Power as a Root and Power

For the positive bases here, an exponent m/n means take an nth root and then an mth power.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Evaluate 8^(2/3) as a real number. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8.
  • Independent check inputs: Evaluate 27^(2/3) as a real number. Candidate answers: (A) 10 (B) 18 (C) 8 (D) 9.

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. For the positive bases here, an exponent m/n means take an nth root and then an mth power. Exact perfect-power inputs make this operation visible without decimal approximation.

Here is the method on the supplied teaching case. Take the cube root first: (2³)^(1/3)=2. Then square: 2²=4. The denominator of the exponent tells which root to take.

The denominator names the root and the numerator names the power. Raising a number to two thirds does not mean multiplying it by two thirds; the number-system and domain assumptions matter.

  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.
Worked example

See the method in action

Teaching case: Evaluate 8^(2/3) as a real number. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8. Solution: Take the cube root first: (2³)^(1/3)=2. Then square: 2²=4. The denominator of the exponent tells which root to take. The final answer is 4. The denominator names the root and the numerator names the power. Raising a number to two thirds does not mean multiplying it by two thirds; the number-system and domain assumptions matter.

Your turn

Check your reasoning

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Helpful tips
  • For the positive bases here, an exponent m/n means take an nth root and then an mth power.
  • The denominator names the root and the numerator names the power.
Evaluate 27^(2/3) as a real number.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

9

Take the cube root first: (3³)^(1/3)=3. Then square: 3²=9. The denominator of the exponent tells which root to take.

Take it further

Changed-input task: Evaluate 729^(2/3) as a real number. Candidate answers: (A) 162 (B) 80 (C) 81 (D) 82. Show the mathematical reason, not only the selected answer. Expected result: 81. Take the cube root first: (9³)^(1/3)=9. Then square: 9²=81. The denominator of the exponent tells which root to take.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Evaluate 64^(2/3) as a real number.
Need a hint?
  • For the positive bases here, an exponent m/n means take an nth root and then an mth power.
  • The denominator names the root and the numerator names the power.
2. Evaluate 125^(2/3) as a real number.
Need a hint?
  • For the positive bases here, an exponent m/n means take an nth root and then an mth power.
  • The denominator names the root and the numerator names the power.
3. Evaluate 216^(2/3) as a real number.
Need a hint?
  • For the positive bases here, an exponent m/n means take an nth root and then an mth power.
  • The denominator names the root and the numerator names the power.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

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These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.