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Grade 8 · lesson 2 of 4

Interpret a Negative Exponent as a Reciprocal

A negative integer exponent represents the reciprocal of a positive power of the same nonzero base.

About 20 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 2^(−2) exactly. Candidate answers: (A) -1/4 (B) -4 (C) 1/4 (D) 4.
  • Independent check inputs: Evaluate 2^(−3) exactly. Candidate answers: (A) -1/8 (B) -8 (C) 1/8 (D) 8.

Learn the idea

Powers encode repeated multiplication. Integer exponent rules preserve that structure, while scientific notation separates a number’s scale from a coefficient that can be compared or operated on efficiently. A negative integer exponent represents the reciprocal of a positive power of the same nonzero base. This extends the quotient pattern without making a positive base’s value negative.

Here is the method on the supplied teaching case. A negative exponent means reciprocal: 2^(−2)=1/(2^2)=1/4. It does not make the value negative.

Keep a negative exponent separate from a negative base or leading minus sign. Zero cannot be used as the base of a negative power because the reciprocal would require division by zero.

  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 2^(−2) exactly. Candidate answers: (A) -1/4 (B) -4 (C) 1/4 (D) 4. Solution: A negative exponent means reciprocal: 2^(−2)=1/(2^2)=1/4. It does not make the value negative. The final answer is 1/4. Keep a negative exponent separate from a negative base or leading minus sign. Zero cannot be used as the base of a negative power because the reciprocal would require division by zero.

Your turn

Check your reasoning

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Helpful tips
  • A negative integer exponent represents the reciprocal of a positive power of the same nonzero base.
  • Keep a negative exponent separate from a negative base or leading minus sign.
Evaluate 2^(−3) exactly.

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See the answer and explanation

1/8

A negative exponent means reciprocal: 2^(−3)=1/(2^3)=1/8. It does not make the value negative.

Take it further

Changed-input task: Evaluate 2^(−9) exactly. Candidate answers: (A) -1/512 (B) -512 (C) 1/512 (D) 512. Show the mathematical reason, not only the selected answer. Expected result: 1/512. A negative exponent means reciprocal: 2^(−9)=1/(2^9)=1/512. It does not make the value negative.

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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 2^(−4) exactly.
Need a hint?
  • A negative integer exponent represents the reciprocal of a positive power of the same nonzero base.
  • Keep a negative exponent separate from a negative base or leading minus sign.
2. Evaluate 2^(−5) exactly.
Need a hint?
  • A negative integer exponent represents the reciprocal of a positive power of the same nonzero base.
  • Keep a negative exponent separate from a negative base or leading minus sign.
3. Evaluate 2^(−6) exactly.
Need a hint?
  • A negative integer exponent represents the reciprocal of a positive power of the same nonzero base.
  • Keep a negative exponent separate from a negative base or leading minus sign.

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

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