maddyhelps

Math 9

Math 9 · Worksheets

Mixed review · Roots, powers and rational numbers

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

All worksheets

Mixed review · Roots, powers and rational numbers

Math 9 · maddyhelps.com

Name
Date
Score
/ 10

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

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

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

  2. −3/4 + 1/2 equals

    1. a) −2/6
    2. b) −5/4
    3. c) 1/4
    4. d) −1/4

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

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

  4. Two cube-shaped boxes, each with 30 cm edges, are taped together face to face. What is the surface area of the pair as one object?

    1. a) 10 800 cm²
    2. b) 9000 cm²
    3. c) 9900 cm²
    4. d) 5400 cm²

  5. √40 lies between which two whole numbers?

    1. a) 7 and 8
    2. b) 5 and 6
    3. c) 6 and 7
    4. d) 20 and 21

  6. A pond's water level dropped 3/8 m each week for 6 weeks. Then a storm raised it 3/4 m. What was the overall change in the water level?

    1. a) −3 m
    2. b) −1½ m
    3. c) 3/8 m
    4. d) 1½ m

  7. Four days' low temperatures in Grande Prairie were −8.5 °C, −3.0 °C, 2.5 °C and −6.0 °C. What was the mean low temperature?

    1. a) −15.0 °C
    2. b) −3.75 °C
    3. c) 3.75 °C
    4. d) −5.0 °C

  8. A cube with 10 cm edges sits in the centre of the top of a rectangular prism that is 20 cm long, 20 cm wide and 5 cm high. What is the surface area of the object?

    1. a) 1800 cm²
    2. b) 1600 cm²
    3. c) 1700 cm²
    4. d) 1300 cm²

  9. The value of −2⁴ is

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

  10. The whole number closest to √85 is

    1. a) 9
    2. b) 10
    3. c) 8
    4. d) 43

Answer key · Mixed review · Roots, powers and rational numbers

Math 9 · maddyhelps.com

  1. c) 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.
  2. d) −1/4 — Use a common denominator: −3/4 + 2/4 = −1/4. The answer is negative because the negative part is bigger. Adding tops and bottoms separately gives −2/6, which is never how fractions add.
  3. b) 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.
  4. b) 9000 cm² — Each face is 30 × 30 = 900 cm², and two cubes have 12 faces: 10 800 cm². Taping them hides one face on each box, so subtract 2 × 900 to get 9000 cm². Subtracting only one face gives 9900.
  5. c) 6 and 7 — Find the perfect squares on either side: 36 < 40 < 49, so √40 is between √36 = 6 and √49 = 7. Halving 40 to get 20 is the most common way to get this wrong.
  6. b) −1½ m — Six drops of 3/8 m is 6 × (−3/8) = −18/8 = −2¼ m. Adding the rise gives −2¼ + ¾ = −1½ m, so the pond ended 1½ m lower. Forgetting to multiply by 6 gives 3/8, and treating the rise as another drop gives −3.
  7. b) −3.75 °C — Add with the signs: −8.5 + (−3.0) + 2.5 + (−6.0) = −15.0, then divide by 4 to get −3.75 °C. Treating 2.5 as −2.5 gives −20.0 ÷ 4 = −5.0 °C; the one day above zero pulls the mean up.
  8. b) 1600 cm² — Cube 6 × 100 = 600 cm², prism 2(400 + 100 + 100) = 1200 cm², and the 100 cm² where they meet is hidden on both: 1800 − 200 = 1600 cm². Only the part of the prism's top under the cube is hidden — removing the whole 400 cm² top gives 1300.
  9. b) −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.
  10. a) 9 — 85 is between 81 = 9² and 100 = 10², and it is only 4 away from 81 but 15 away from 100, so √85 is just over 9 (about 9.2). 43 comes from halving 85, which is not what a square root does.