Answer :
Marie made her first error in Step 1. Let's break down the correct solution step-by-step:
### Step-by-Step Solution:
1. Given Expression: \(\left(x^{-3} y^2 \cdot x\right)^7\)
2. Combine the Powers of \(x\):
- The expression inside the parentheses needs to be simplified first.
- \(x^{-3} \cdot x = x^{-3 + 1} = x^{-2}\)
Therefore, the expression inside the parentheses becomes:
[tex]\[(x^{-2} y^2)\][/tex]
3. Apply the Exponent to Each Factor:
- Now, raise each term inside the parentheses to the power of 7.
- \((x^{-2})^7 = x^{-2 \cdot 7} = x^{-14}\)
- \((y^2)^7 = y^{2 \cdot 7} = y^{14}\)
So the expression becomes:
[tex]\[x^{-14} y^{14}\][/tex]
4. Combine the Powers:
- The simplified expression is now:
[tex]\[x^{-14} y^{14}\][/tex]
5. Final Answer:
- The given expression simplifies to \(x^{-14} y^{14}\).
- Written in standard form, \(x^{-14} y^{14}\) simply means:
[tex]\[ \frac{y^{14}}{x^{14}} \][/tex]
Therefore, the correct simplified version of the given expression is:
[tex]\[ \left(x^{-3} y^2 \cdot x\right)^7 = \frac{y^{14}}{x^{14}} \][/tex]
Thus, Marie's first error was in Step 1, where she incorrectly wrote [tex]\(\left(x^3 y^2 \cdot x\right)^7\)[/tex] instead of correctly simplifying the power of [tex]\(x\)[/tex] first as [tex]\(\left(x^{-2} y^2\right)^7\)[/tex].
### Step-by-Step Solution:
1. Given Expression: \(\left(x^{-3} y^2 \cdot x\right)^7\)
2. Combine the Powers of \(x\):
- The expression inside the parentheses needs to be simplified first.
- \(x^{-3} \cdot x = x^{-3 + 1} = x^{-2}\)
Therefore, the expression inside the parentheses becomes:
[tex]\[(x^{-2} y^2)\][/tex]
3. Apply the Exponent to Each Factor:
- Now, raise each term inside the parentheses to the power of 7.
- \((x^{-2})^7 = x^{-2 \cdot 7} = x^{-14}\)
- \((y^2)^7 = y^{2 \cdot 7} = y^{14}\)
So the expression becomes:
[tex]\[x^{-14} y^{14}\][/tex]
4. Combine the Powers:
- The simplified expression is now:
[tex]\[x^{-14} y^{14}\][/tex]
5. Final Answer:
- The given expression simplifies to \(x^{-14} y^{14}\).
- Written in standard form, \(x^{-14} y^{14}\) simply means:
[tex]\[ \frac{y^{14}}{x^{14}} \][/tex]
Therefore, the correct simplified version of the given expression is:
[tex]\[ \left(x^{-3} y^2 \cdot x\right)^7 = \frac{y^{14}}{x^{14}} \][/tex]
Thus, Marie's first error was in Step 1, where she incorrectly wrote [tex]\(\left(x^3 y^2 \cdot x\right)^7\)[/tex] instead of correctly simplifying the power of [tex]\(x\)[/tex] first as [tex]\(\left(x^{-2} y^2\right)^7\)[/tex].
