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Math 10C

Math 10C · Worksheets

Factors, roots and powers

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

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Factors, roots and powers

Math 10C · maddyhelps.com

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  1. The prime factorization of 60 is

    1. a) 2² × 3 × 5
    2. b) 2 × 30
    3. c) 4 × 15
    4. d) 2 × 3 × 10

  2. Of these numbers, the irrational one is

    1. a) 0.75
    2. b) ⅔
    3. c) √7
    4. d) √9

  3. In simplest form, √48 is

    1. a) 2√12
    2. b) 4√12
    3. c) 16√3
    4. d) 4√3

  4. Simplified, (2x³y)⁴ is

    1. a) 16x¹²y⁴
    2. b) 16x⁷y
    3. c) 8x⁷y⁴
    4. d) 2x¹²y⁴

  5. The least common multiple of 8 and 12 is

    1. a) 96
    2. b) 48
    3. c) 24
    4. d) 4

  6. The value of 16^(1/2) is

    1. a) 32
    2. b) 4
    3. c) 2
    4. d) 8

  7. The value of 3⁻² is

    1. a) 9
    2. b) −6
    3. c) 1/9
    4. d) −9

  8. x⁵ × x³ equals

    1. a) x⁸
    2. b) x¹⁵
    3. c) 2x⁸
    4. d) x²

  9. The sum of a rational number and an irrational number is

    1. a) always a whole number
    2. b) always rational
    3. c) always irrational
    4. d) sometimes rational

  10. The greatest common factor of 24 and 36 is

    1. a) 72
    2. b) 12
    3. c) 6
    4. d) 4

Answer key · Factors, roots and powers

Math 10C · maddyhelps.com

  1. a) 2² × 3 × 5 — Keep splitting until everything is prime: 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5. The other options all still contain composite numbers, so they are factorizations but not prime ones.
  2. c) √7 — √9 is exactly 3, and 0.75 and ⅔ are both ratios of whole numbers, so all three are rational. √7 has a decimal that never ends and never repeats, so it cannot be written as a fraction.
  3. d) 4√3 — Pull out the largest perfect square factor: 48 = 16 × 3, so √48 = 4√3. The option 2√12 is equal in value but not simplified, because 12 still contains the perfect square 4.
  4. a) 16x¹²y⁴ — The outside exponent applies to every factor inside: 2⁴ = 16, (x³)⁴ = x¹², and y⁴. Forgetting to raise the coefficient is the most common error, and it gives 2x¹²y⁴.
  5. c) 24 — 8 = 2³ and 12 = 2² × 3, so the LCM takes the highest power of each prime: 2³ × 3 = 24. Multiplying the two numbers gives 96, which is a common multiple but not the least one.
  6. b) 4 — A fractional exponent is a root: the denominator says which root. So 16^(1/2) = √16 = 4. And 16^(3/2) would be (√16)³ = 64 — take the root first, then the power, because the numbers stay smaller.
  7. c) 1/9 — A negative exponent means the reciprocal, not a negative answer: 3⁻² = 1/3² = 1/9. The sign of the exponent controls which side of the fraction line the power sits on.
  8. a) x⁸ — Multiplying powers of the same base adds the exponents, because you are just counting how many copies of x there are in total: five plus three is eight. The bases do not multiply.
  9. c) always irrational — If the sum were rational, subtracting the rational part would make the irrational number rational too, which is a contradiction. Note the contrast: the sum of two <i>irrational</i> numbers can be rational, as √2 + (−√2) shows.
  10. b) 12 — Compare prime factorizations: 24 = 2³ × 3 and 36 = 2² × 3², so the common part is 2² × 3 = 12. Taking the <i>lowest</i> power of each shared prime gives the GCF; the highest gives the LCM.