Question

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

Answer 1

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**.