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Negative bases and exponents

See how parentheses change powers of negative numbers.

Practice + answer key · 2 pages

Worksheet settings12 questions · 2 columns · A4
Base magnitude

Choose the positive magnitude of each negative base; exponents range from 2 to 5.

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mathsprig-1
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1720f434
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1 page
Negative bases and exponentsNegative bases and exponents. Practice. Evaluate −4^2. Evaluate −6^5. Evaluate (−4)^5. Evaluate (−3)^4. Evaluate (−5)^3. Evaluate (−4)^3. Evaluate (−3)^5. Evaluate −4^4. Evaluate −5^5. Evaluate −2^5. Evaluate (−3)^4. Evaluate −3^3.Name __________________________Date _______________Negative bases and exponentsEvaluate each expression, paying attention to parentheses.Worked exampleEvaluate (−3)² and −3². → 9 and −9(−3)² = (−3) × (−3) = 9.1.Evaluate −4^2.2.Evaluate −6^5.3.Evaluate (−4)^5.4.Evaluate (−3)^4.5.Evaluate (−5)^3.6.Evaluate (−4)^3.7.Evaluate (−3)^5.8.Evaluate −4^4.9.Evaluate −5^5.10.Evaluate −2^5.11.Evaluate (−3)^4.12.Evaluate −3^3.MathSprig · mathsprig.site · support@mathsprig.sitePage 1 of 1mathsprig-1 · mathsprig-v1 · 1720f434
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Practice + answer key · 2 pages

Integers

What you’ll practice

Evaluate powers with a negative base or an external negative sign.

Compare (−3)² with −3². In the first, the negative sign is part of the base; in the second, you square 3 before applying the negative sign.

How this skill works

In (−a)ⁿ, multiply the complete negative base n times. An even exponent gives a positive result and an odd exponent gives a negative result.

In −aⁿ, evaluate aⁿ first, then apply the negative sign.

Worked example

Evaluate (−3)² and −3².
  1. (−3)² = (−3) × (−3) = 9.
  2. −3² = −(3 × 3) = −9.

Answer: 9 and −9

Teaching notes

  • Do not move a negative sign inside parentheses unless the expression allows it.
Practice questions
  1. Evaluate −4^2.
  2. Evaluate −6^5.
  3. Evaluate (−4)^5.
  4. Evaluate (−3)^4.
  5. Evaluate (−5)^3.
  6. Evaluate (−4)^3.
  7. Evaluate (−3)^5.
  8. Evaluate −4^4.
  9. Evaluate −5^5.
  10. Evaluate −2^5.
  11. Evaluate (−3)^4.
  12. Evaluate −3^3.
Answers and steps
  1. −16
    1. The exponent applies to 4; the negative sign is applied afterward.
    2. −4^2 = −16.
  2. −7776
    1. The exponent applies to 6; the negative sign is applied afterward.
    2. −6^5 = −7776.
  3. −1024
    1. The parentheses include the negative sign in the base. There are 5 factors of −4.
    2. (−4)^5 = −1024.
  4. 81
    1. The parentheses include the negative sign in the base. There are 4 factors of −3.
    2. (−3)^4 = 81.
  5. −125
    1. The parentheses include the negative sign in the base. There are 3 factors of −5.
    2. (−5)^3 = −125.
  6. −64
    1. The parentheses include the negative sign in the base. There are 3 factors of −4.
    2. (−4)^3 = −64.
  7. −243
    1. The parentheses include the negative sign in the base. There are 5 factors of −3.
    2. (−3)^5 = −243.
  8. −256
    1. The exponent applies to 4; the negative sign is applied afterward.
    2. −4^4 = −256.
  9. −3125
    1. The exponent applies to 5; the negative sign is applied afterward.
    2. −5^5 = −3125.
  10. −32
    1. The exponent applies to 2; the negative sign is applied afterward.
    2. −2^5 = −32.
  11. 81
    1. The parentheses include the negative sign in the base. There are 4 factors of −3.
    2. (−3)^4 = 81.
  12. −27
    1. The exponent applies to 3; the negative sign is applied afterward.
    2. −3^3 = −27.