Welcome to Westonci.ca, the ultimate question and answer platform. Get expert answers to your questions quickly and accurately. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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].
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.