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Grade 9 · Math · Algebra 1

9th grade Rational exponents and the laws of exponents worksheet

6 printable versionsChallenge Me10 questions · answer keyFree · no account

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Rational exponents and the laws of exponents

Grade 9 Math Worksheet - Challenge Me

NameDate

Circle the best answer.

  1. A town's population multiplies by 81 over 4 years at a steady yearly factor f. Kira wants f. What is it?

    • 21
    • 20.25
    • 3
    • 9

  2. A video's views multiply by 8 over 3 weeks at a steady weekly factor f. Luz wants f. What is it?

    • about 2.67
    • about 3.33
    • about 2.83
    • 2

  3. An account multiplies by 25 in one year at a steady rate. Jun wants the factor for each half-year. What is it?

    • 12.5
    • 13
    • about 2.9
    • 5

  4. Noah writes 9^(1/2) + 16^(1/2) = 25^(1/2) = 5. Which response is correct?

    • No: the exponents add too, so 9^(1/2) + 16^(1/2) = 25^1 = 25.
    • No: a half power halves the base, so 9^(1/2) + 16^(1/2) = 4.5 + 8 = 12.5.
    • No: 9^(1/2) + 16^(1/2) = 3 + 4 = 7, but 25^(1/2) = 5, so the sum cannot be combined.
    • Yes: powers with the same exponent add their bases, so the sum is 25^(1/2) = 5.

  5. Milo writes (-4)^(1/2) = -2, because the square root of 4 is 2 and the minus sign comes along. Which response is correct?

    • Yes: (-2) × (-2) = -4, so -2 squared is -4.
    • Yes: the square root of 4 is 2, and the minus sign stays, so it is -2.
    • No: (-2)² = 4, not -4, and no real number squares to -4.
    • No: the minus sign drops under a root, so (-4)^(1/2) = 2.

  6. Tala writes (2^(1/3))^6 = 2^(1/18), because the exponents multiply. Which response is correct?

    • No: six copies add, so (2^(1/3))^6 = 6 × 2^(1/3), about 7.6.
    • No: the exponents add, so (2^(1/3))^6 = 2^(19/3).
    • Yes: the 6 goes under the 3, so the exponent is 1/18 and the value is 2^(1/18).
    • No: 1/3 × 6 = 2, so (2^(1/3))^6 = 2² = 4.

  7. Ren solves (x - 2)^(1/2) = 6 for x. What is x?

    • 34
    • 38
    • 36
    • about 4.4

  8. Bo solves (x + 5)^(1/3) = 2 for x. What is x?

    • 3
    • -1
    • 8
    • 13

  9. Esme solves (x + 3)^(1/2) = 5 for x. What is x?

    • 28
    • 22
    • 25
    • 7

  10. Zee solves 2 × x^(1/3) = 8 for x. What is x?

    • 4
    • 512
    • 64
    • 16
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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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