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Powers and exponent laws

Ten questions of mixed difficulty, covering Powers. Print it, or work through it on screen — the answer key starts on its own page.

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Powers and exponent laws

Math 9 · maddyhelps.com

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Circle the best answer for each question. Show your work in the space provided.

  1. Written as a single power, 5³ × 5⁴ is

    1. a) 5⁷
    2. b) 5¹²
    3. c) 25⁷
    4. d) 5¹

  2. The value of (−2)³ − 3² is

    1. a) −17
    2. b) −1
    3. c) 1
    4. d) −12

  3. Written as a single power, 6⁸ ÷ 6² is

    1. a) 6¹⁰
    2. b) 6⁶
    3. c) 1⁶
    4. d) 6⁴

  4. The value of (2⁵ × 2³) ÷ (2²)³ is

    1. a) 512
    2. b) 1024
    3. c) 8
    4. d) 4

  5. The value of 12 − (−6/3)² is

    1. a) 14
    2. b) 0
    3. c) 16
    4. d) 8

  6. A cube-shaped box has edges 4² cm long. Its volume, written as a single power of 4, is

    1. a) 4⁶ cm³
    2. b) 12² cm³
    3. c) 4⁵ cm³
    4. d) 4⁸ cm³

  7. The value of (−2 × 3)² − 2 × 3² is

    1. a) 0
    2. b) −54
    3. c) −36
    4. d) 18

  8. The value of −2⁴ is

    1. a) 16
    2. b) 8
    3. c) −8
    4. d) −16

  9. The value of 9⁰ is

    1. a) 9
    2. b) 0
    3. c) undefined
    4. d) 1

  10. The value of (−3)⁴ is

    1. a) 81
    2. b) 12
    3. c) −81
    4. d) −12

Answer key · Powers and exponent laws

Math 9 · maddyhelps.com

  1. a) 5⁷ — Multiplying powers with the same base adds the exponents, because you are counting factors of 5: three and then four more make seven. Multiplying the exponents gives 5¹², and multiplying the bases gives 25⁷ — the base stays 5.
  2. a) −17 — (−2)³ = −8, because three negative factors give a negative, and 3² = 9, so −8 − 9 = −17. Treating the minus sign as part of the base turns − 3² into + 9 and gives 1, but the square belongs to the 3 alone.
  3. b) 6⁶ — Dividing powers with the same base subtracts the exponents: 8 − 2 = 6. Dividing the exponents gives 6⁴, and dividing the bases gives 1⁶. Writing out eight 6s and cancelling two shows why subtracting is right.
  4. d) 4 — Top: 2⁵ × 2³ = 2⁸ (add the exponents). Bottom: (2²)³ = 2⁶ (multiply them). Then 2⁸ ÷ 2⁶ = 2² = 4; mixing up the two laws — adding in (2²)³, or multiplying in 2⁵ × 2³ — gives 8 or 512.
  5. d) 8 — Inside the bracket, −6/3 = −2, and (−2)² = 4, so 12 − 4 = 8. Squaring only the top gives 36/3 = 12 and an answer of 0 — a power of a quotient squares the top and the bottom. Making (−2)² negative gives 16.
  6. a) 4⁶ cm³ — Volume is edge³, so (4²)³. A power of a power multiplies the exponents: 2 × 3 = 6, giving 4⁶ cm³. Adding them gives 4⁵, and cubing the exponent (2³ = 8) gives 4⁸ — but three groups of two 4s is six 4s.
  7. d) 18 — The bracket is squared as a whole: (−6)² = 36. In 2 × 3² only the 3 is squared: 2 × 9 = 18, so 36 − 18 = 18. Squaring 2 × 3 together in the second term gives 36 − 36 = 0; without brackets, the power belongs to the 3 alone.
  8. d) −16 — With no brackets, the exponent applies only to the 2, and the negative is applied afterwards: −2⁴ = −(2 × 2 × 2 × 2) = −16. Reading it as (−2)⁴ gives 16, the single most common error with powers.
  9. d) 1 — Follow the pattern 9² = 81, 9¹ = 9: each step down in the exponent divides by 9, so 9⁰ = 9 ÷ 9 = 1. Any base except 0 raised to the exponent 0 is 1, not 0.
  10. a) 81 — (−3)⁴ = (−3)(−3)(−3)(−3). Four negative factors pair up into positives, so the answer is 81. −12 and 12 come from multiplying the base by the exponent, which is not what a power means.