Exponent Rules practice questions

Exponent Rules questions

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1. 1Exponent rulesFoundation

   Which expression is equivalent to (2x³)(5x⁴)?Which expression is equivalent to (2x^3)(5x^4)?

   • A7x¹²7x^12
   • B10x⁷10x^7
   • C7x⁷7x^7
   • D10x¹²10x^12
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   Answer: 10x⁷10x^7

   Multiply the coefficients: 2 × 5 = 10. Add the exponents on the shared base: x³ x⁴ = x⁷.Multiply the coefficients: 2 × 5 = 10. Add the exponents on the shared base: x^3 x^4 = x^7.

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2. 2Exponent rulesFoundation

   If x is not zero, which expression is equivalent to 3x⁴3x⁴?If x is not zero, which expression is equivalent to (3x^4)/(3x^4)?

   • A11
   • B00
   • Cxx
   • D33
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   Answer: 11

   Any nonzero quantity divided by itself is 1. Using the quotient rule gives x^4-4 = x⁰, and x⁰ = 1 for every nonzero x.Any nonzero quantity divided by itself is 1. Using the quotient rule gives x^(4-4) = x^0, and x^0 = 1 for every nonzero x.

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3. 3Exponent rulesStandard

   What is the value of 5^-2?What is the value of 5^(-2)?

   • A-25-25
   • B1251/25
   • C-10-10
   • D2525
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   Answer: 1251/25

   A negative exponent means the reciprocal of the positive power: 5^-2 = 15^2 = 125. It does not make the result negative.A negative exponent means the reciprocal of the positive power: 5^(-2) = 1/5^2 = 1/25. It does not make the result negative.

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4. 4Exponent rulesStandard

   Which expression is equivalent to (8x⁶)^(13)?Which expression is equivalent to (8x^6)^(1/3)?

   • A2x³2x^3
   • B8x²8x^2
   • C2x²2x^2
   • D4x²4x^2
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   Answer: 2x²2x^2

   A cube root applies to both factors: the cube root of 8 is 2, and x⁶ raised to 13 gives x² because the exponents multiply.A cube root applies to both factors: the cube root of 8 is 2, and x^6 raised to 1/3 gives x^2 because the exponents multiply.

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5. 5Exponent rulesStandard

   Which expression is equivalent to x⁵ y²x² y⁵, for x and y not zero?Which expression is equivalent to (x^5 y^2)/(x^2 y^5), for x and y not zero?

   • Ax⁷ / y⁷x^7 / y^7
   • B1x³ y³1/(x^3 y^3)
   • Cx³ y³x^3 y^3
   • Dx³ / y³x^3 / y^3
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   Answer: x³ / y³x^3 / y^3

   Subtract exponents on each base: x^5-2 = x³ and y^2-5 = y^-3, which is 1/y³. Together that is x³ / y³.Subtract exponents on each base: x^(5-2) = x^3 and y^(2-5) = y^(-3), which is 1/y^3. Together that is x^3 / y^3.

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6. 6Exponent rulesStandard

   Which expression is equivalent to (2x³ y)^44x⁵ y², where x and y are positive?Which expression is equivalent to (2x^3 y)^4/(4x^5 y^2), where x and y are positive?

   • A8x⁷ y²8x^7 y^2
   • B4x¹² y²4x^12 y^2
   • C2x⁷ y²2x^7 y^2
   • D4x⁷ y²4x^7 y^2
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   Answer: 4x⁷ y²4x^7 y^2

   The power of 4 applies to every factor: (2x³ y)^4 = 16x¹² y⁴. Dividing by 4x⁵ y² gives 164 = 4, x^12-5 = x⁷ and y^4-2 = y². Leaving the exponent on x undivided gives x¹²; forgetting to raise the 2 gives the wrong coefficient.The power of 4 applies to every factor: (2x^3 y)^4 = 16x^12 y^4. Dividing by 4x^5 y^2 gives 16/4 = 4, x^(12-5) = x^7 and y^(4-2) = y^2. Leaving the exponent on x undivided gives x^12; forgetting to raise the 2 gives the wrong coefficient.

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7. 7Exponent rulesStandard

   If 3^a multiplied by 3⁵ equals 3¹², enter the value of a.If 3^a multiplied by 3^5 equals 3^12, enter the value of a.

   Student-produced response: write your own answer rather than choosing one.

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   Answer: 77

   Multiplying powers of the same base adds the exponents, so a + 5 = 12 and a = 7.Multiplying powers of the same base adds the exponents, so a + 5 = 12 and a = 7.

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8. 8Exponent rulesStandard

   For positive x and y, which expression is equivalent to x⁵ y²x² y⁵?For positive x and y, which expression is equivalent to (x^5 y^2)/(x^2 y^5)?

   • Ax³ y³x^3 y^3
   • Bx³ / y³x^3 / y^3
   • Cy³ / x³y^3 / x^3
   • Dx⁷ y⁷x^7 y^7
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   Answer: x³ / y³x^3 / y^3

   Dividing subtracts exponents: x^5-2 = x³ and y^2-5 = y^-3, and a negative exponent means the term moves to the denominator. Multiplying the exponents instead of subtracting gives the last choice.Dividing subtracts exponents: x^(5-2) = x^3 and y^(2-5) = y^(-3), and a negative exponent means the term moves to the denominator. Multiplying the exponents instead of subtracting gives the last choice.

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9. 9Exponent rulesStandard

   Which expression is equivalent to (2a³)^3?Which expression is equivalent to (2a^3)^3?

   • A6a⁹6a^9
   • B8a⁹8a^9
   • C2a⁹2a^9
   • D8a⁶8a^6
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   Answer: 8a⁹8a^9

   The power of 3 applies to both factors: 2³ = 8 and (a³)^3 = a⁹. Multiplying the coefficient by the exponent gives 6a⁹, and adding the exponents instead of multiplying gives a⁶.The power of 3 applies to both factors: 2^3 = 8 and (a^3)^3 = a^9. Multiplying the coefficient by the exponent gives 6a^9, and adding the exponents instead of multiplying gives a^6.

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10. 10Exponent rulesStandard

    Which expression is equivalent to (3m² n)^2?Which expression is equivalent to (3m^2 n)^2?

    • A6m⁴ n²6m^4 n^2
    • B9m⁴ n²9m^4 n^2
    • C9m⁴ n9m^4 n
    • D3m⁴ n²3m^4 n^2
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    Answer: 9m⁴ n²9m^4 n^2

    The outer square applies to every factor: 3² = 9, (m²)^2 = m⁴ and n². Multiplying the coefficient by the exponent gives 6, and leaving n unsquared drops one factor.The outer square applies to every factor: 3^2 = 9, (m^2)^2 = m^4 and n^2. Multiplying the coefficient by the exponent gives 6, and leaving n unsquared drops one factor.

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11. 11Exponent rulesStandard

    Which expression is equivalent to (4p³ q²)^2?Which expression is equivalent to (4p^3 q^2)^2?

    • A8p⁶ q⁴8p^6 q^4
    • B16p⁵ q⁴16p^5 q^4
    • C16p⁹ q⁴16p^9 q^4
    • D16p⁶ q⁴16p^6 q^4
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    Answer: 16p⁶ q⁴16p^6 q^4

    The outer square applies to every factor: 4² = 16, (p³)^2 = p⁶ and (q²)^2 = q⁴. Adding the exponents rather than multiplying gives p⁵.The outer square applies to every factor: 4^2 = 16, (p^3)^2 = p^6 and (q^2)^2 = q^4. Adding the exponents rather than multiplying gives p^5.

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12. 12Exponent rulesChallenge

    If 4^x = 8, what is the value of x?If 4^x = 8, what is the value of x?

    • A1.51.5
    • B33
    • C0.50.5
    • D22
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    Answer: 1.51.5

    Write both sides on base 2: 4^x = 2^2x and 8 = 2³. So 2x = 3 and x = 1.5. Matching bases is the move that makes this a one-line problem.Write both sides on base 2: 4^x = 2^(2x) and 8 = 2^3. So 2x = 3 and x = 1.5. Matching bases is the move that makes this a one-line problem.

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