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Grade 9 · Math · Algebra 1

9th grade Rational exponents and the laws of exponents worksheet

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Rational exponents and the laws of exponents

Grade 9 Math Worksheet

NameDate

Circle the best answer.

  1. Kira writes (x^6)^(1/2) as one power of x, for x > 0. Which is it?

    • x³
    • x^(1/6)
    • x^(13/2)
    • x^12

  2. A medicine's amount drops to 1/4 each hour: A = 80 × (1/4)^t, with t in hours. Pia finds A at t = 1/2. How many mg remain?

    • 50
    • 40
    • 5
    • 10

  3. Bacteria multiply by 16 every day at a steady rate. Noah wants the factor for one quarter of a day. By what factor do they multiply?

    • 65,536
    • 2
    • 4.75
    • 4

  4. A sample keeps 1/8 of its mass each week: M = 96 × (1/8)^w, with w in weeks. Zee finds M at w = 1/3. How many grams remain?

    • 68
    • 4
    • 48
    • about 0.19

  5. A cube-shaped box's volume becomes 64 times as big. Bo wants the factor for each edge. By what factor is each edge multiplied?

    • 4
    • 8
    • 64
    • about 21.3

  6. Tala's cube-shaped planter has edge 30 cm. A bigger similar cube holds 8 times as much soil. What is the new edge?

    • about 84.9 cm
    • 60 cm
    • 80 cm
    • 240 cm

  7. A rectangular poster is reduced to 1/4 of its area, keeping its shape. Its width was 40 cm. Milo finds the new width. What is it?

    • 5 cm
    • about 25.2 cm
    • 10 cm
    • 20 cm

  8. Pia's robot screen shows 25^(-1/2), a power with a minus sign. What is its value?

    • -5
    • 5
    • 1/5
    • 2/25

  9. Noah compares 4^(-1/2) and 27^(-1/3). Which statement is correct?

    • 4^(-1/2) is greater: it equals -2, and 27^(-1/3) equals -3.
    • 27^(-1/3) is greater, because 27 is a larger base than 4.
    • 27^(-1/3) is greater: it equals 3, and 4^(-1/2) equals 2.
    • 4^(-1/2) is greater: it equals 1/2, and 27^(-1/3) equals 1/3.

  10. Esme compares 9^(-1/2) and 125^(-1/3). Which statement is correct?

    • 125^(-1/3) is greater: it equals 5, and 9^(-1/2) equals 3.
    • 9^(-1/2) is greater: it equals -3, and 125^(-1/3) equals -5.
    • 9^(-1/2) is greater: it equals 1/3, and 125^(-1/3) equals 1/5.
    • 125^(-1/3) is greater, because 125 is a larger base than 9.
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9th grade Rational exponents and the laws of exponents worksheet – printable
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About this worksheet

Rational exponents and the laws of exponents. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can say why 25^(1/2) is 5: squaring it must give 25^1, and the power law says the exponents multiply.
  • I can simplify (x^6)^(1/2) and (27x^9)^(1/3) by multiplying exponents and taking the root of the number.
  • I can turn a growth factor into the factor for part of a period: 9-fold an hour is 3-fold every half hour.
  • I can solve (x + 3)^(1/2) = 5 by squaring both sides, then subtracting.

Standard: N-RN.A.1, N-RN.A.2

Words to know

exponent: The small raised number; it says how a number is used as a factor.

unit fraction exponent: An exponent with 1 on top, like 1/2 or 1/3. It names a root.

equal factors: Numbers that are the same and multiply to the whole: 27 = 3 × 3 × 3.

steady factor: The number a quantity is multiplied by in each period, like ×3 each year.

base: The number or letter that the exponent works on: 5 in 5^(1/2).

Common mistakes

Your child read a fraction power as multiplying the base by the fraction (64 × 1/3).

Your child turned the power 1/k into the power k (25^(1/2) into 25²).

Your child gave plus and minus for a half power.

What to say

Ask: which number times itself, three times, makes 64?

Ask: does a power of 1/2 make 25 bigger or smaller?

Ask: does 25^(1/2) name the positive number that squares to 25, or both?

Questions parents ask

What is the most common mistake with rational exponents and the laws of exponents?

Your child read a fraction power as multiplying the base by the fraction (64 × 1/3).

How can I help my child with rational exponents and the laws of exponents?

Ask: which number times itself, three times, makes 64?

Are these 9th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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