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Rena used the steps below to evaluate the expression (StartFraction (x Superscript negative 3 Baseline) (y Superscript negative 2 Baseline) Over 2 (x Superscript 4 Baseline) (y superscript negative 4 Baseline) EndFraction) Superscript negative 3, when x = negative 1 and y = 2.

Step 1:
Substitute x = negative 1 and y = 2 into the expression.
(StartFraction (negative 1) Superscript negative 3 Baseline (2) Superscript negative 2 Baseline Over 2 (negative 1) Superscript 4 Baseline (2) superscript negative 4 Baseline) EndFraction) Superscript negative 3
Step 2:
Simplify the parentheses.
(StartFraction (2) Superscript 4 Baseline Over 2 (negative 1) Superscript 4 Baseline (negative 1) cubed (2) squared EndFraction) Superscript negative 3 Baseline = (StartFraction (2) squared Over 2 (negative 1) Superscript 7 Baseline EndFraction) Superscript negative 3
Step 3:
Evaluate the power to a power.
StartFraction (2) Superscript negative 6 Baseline Over 2 Superscript negative 3 Baseline (negative 1) Superscript negative 21 baseline EndFraction
Step 4:
Use reciprocals and find the value.
StartFraction 1 Over 2 cubed (2) Superscript 6 Baseline (negative 1) Superscript 21 Baseline EndFraction = StartFraction 1 Over 8 times 64 times (negative 1) EndFraction = Negative StartFraction 1 Over 512 EndFraction

In which step did Rena make the first error?
Step 1
Step 2
Step 3
Step 4

1 answer

Let's go through Rena's steps one by one to pinpoint the error:

Step 1: Rena correctly substituted \( x = -1 \) and \( y = 2 \) into the expression: \[ \left( \frac{(-1)^{-3} (2)^{-2}}{2 (-1)^{4} (2)^{-4}} \right)^{-3} \] This step appears to be fine.

Step 2: Now let's simplify the expression inside the parentheses:

  1. Calculate each part:
    • \( (-1)^{-3} = -1 \)
    • \( (2)^{-2} = \frac{1}{4} \)
    • \( (-1)^4 = 1 \)
    • \( (2)^{-4} = \frac{1}{16} \)

Now substituting these values back into the fraction: \[ \left( \frac{-1 \cdot \frac{1}{4}}{2 \cdot 1 \cdot \frac{1}{16}} \right)^{-3} \] Simplifying the numerator: \[

  • \frac{1}{4} \] And the denominator: \[ 2 \cdot \frac{1}{16} = \frac{2}{16} = \frac{1}{8} \] So we have: \[ \frac{-\frac{1}{4}}{\frac{1}{8}} = -\frac{1}{4} \cdot 8 = -2 \] Thus the expression simplifies to: \[ (-2)^{-3} \]

Rena incorrectly simplified in this step, making an error when simplifying fractions. The correct simplification is not what she arrived at. Therefore, Rena made her first error in Step 2.

Conclusion: The first error Rena made is in Step 2.