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Two students have partially simplified the same expression.

Audrey’s work:

(
x3y−2
22x−5y3
) 3 = (
x8y−5
22
) 3 =
x24y−15
26

David’s work:

(
x3y−2
22x−5y3
) 3 =
x9y−6
26x−15y9

Which statements are true? Check all that apply.
David correctly simplified the power of products in his first step by adding exponents.
Audrey first applied division of like bases by subtracting the exponents.
Next David will apply the quotient of powers.
David’s work is incorrect.
Audrey’s work is correct.
The correct simplified final answer isStartFraction x Superscript 24 Baseline Over 64 y Superscript 15 Baseline EndFraction.

1 answer

Let's analyze the work of both students step by step.

Audrey's Work:

  1. Starting Expression: \[ \left( \frac{x^3 y^{-2}}{2^{2} x^{-5} y^{3}} \right)^3 \]

  2. Simplifying the Division:

    • For the \(x\) terms: \(x^{3 - (-5)} = x^{3 + 5} = x^{8}\)
    • For the \(y\) terms: \(y^{-2 - 3} = y^{-2 - 3} = y^{-5}\)
    • For the constants: \(\frac{1}{2^2} = \frac{1}{4} = \frac{1}{2^2}\)

    So the expression simplifies to: \[ \left( \frac{x^{8} y^{-5}}{2^2} \right)^3 \]

  3. Applying the Power Rule:

    • \(x^{8 \times 3} = x^{24}\)
    • \(y^{-5 \times 3} = y^{-15}\)
    • \(2^{2 \times 3} = 2^6\)

    Thus: \[ \frac{x^{24} y^{-15}}{2^6} \]

David's Work:

  1. Starting Expression: \[ \left( \frac{x^3 y^{-2}}{2^{2} x^{-5} y^{3}} \right)^3 \]

  2. Simplifying the Division:

    • For the \(x\) terms: \(x^{3 - (-5)} = x^{3 + 5} = x^{8}\)
    • For the \(y\) terms: \(y^{-2 - 3} = y^{-2 - 3} = y^{-5}\)
    • The constant terms are similar.

    However, David claims: \[ \frac{x^9 y^{-6}}{26 x^{-15} y^{9}}, \] which is incorrect because he does not simplify correctly.

Evaluating the Statements:

  1. David correctly simplified the power of products in his first step by adding exponents.

    • False. David made an error in his calculations.

  2. Audrey first applied division of like bases by subtracting the exponents.

    • True. Audrey correctly subtracted exponents for both \(x\) and \(y\) while simplifying.

  3. Next David will apply the quotient of powers.

    • False. David has already made errors in his initial steps.

  4. David’s work is incorrect.

    • True. David's work contains errors in the calculations.

  5. Audrey’s work is correct.

    • True. Audrey's process and final answer for her work are correct:
      \[ \frac{x^{24}}{2^6 y^{15}} = \frac{x^{24}}{64 y^{15}}. \]

  6. The correct simplified final answer is \( \frac{x^{24}}{64 y^{15}} \).

    • True. This is the correct simplified final answer based on Audrey's work.

Summary of True Statements:

  • Audrey first applied division of like bases by subtracting the exponents.
  • David’s work is incorrect.
  • Audrey’s work is correct.
  • The correct simplified final answer is \( \frac{x^{24}}{64 y^{15}} \).