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The question is asking whether the expressions \( (-x)^y \) and \( -x^y \) have the same value, and which statement correctly identifies if it's true or false with the appropriate justification.

Let's break this down:

1. **Expression \( (-x)^y \)**: Here, the base is \( -x \), meaning that the negative sign is included with \( x \) before being raised to the power of \( y \). This means that whether the result is positive or negative depends on whether \( y \) is even or odd.
   - If \( y \) is even, \( (-x)^y \) will result in a positive number because raising a negative number to an even power results in a positive outcome.
   - If \( y \) is odd, \( (-x)^y \) will result in a negative number because raising a negative number to an odd power results in a negative outcome.

2. **Expression \( -x^y \)**: This expression means \( x \) is raised to the power \( y \), and then the negative sign is applied. This means that the negative sign is **outside** the exponentiation.
   - No matter the value of \( y \), the result of \( x^y \) is positive (if \( y \) is even) or negative (if \( y \) is odd), but the final result is **always negative** because the negative sign is applied afterward.

Thus, **these two expressions are not the same**. The first expression applies the exponent to the entire base, including the negative sign, while the second applies the exponent only to \( x \), and then applies the negative sign.

- **Correct Answer**: B) false; The base \( -x \) is used as a factor \( y \) times. The expression \( -x^y \) means the opposite of \( x^y \).

Would you like more details on the reasoning, or do you have any other questions? Here are five related questions to deepen your understanding:

1. What happens when you raise a negative number to an odd power?
2. How does exponentiation affect the sign of a number when the base is negative?
3. Can you explain why \( (-x)^2 \) is positive while \( -x^2 \) is negative?
4. What are the rules for handling powers and signs when working with negative bases?
5. What is the difference between \( -x^y \) and \( -(x^y) \) in terms of evaluation?

**Tip**: Always pay close attention to how parentheses are used in expressions involving exponents and negative numbers. They can change the entire meaning of the expression.

The expressions (-x)^y and -x^y have the same value. Which statement correctly identifies if it's true or false and provides the correct justification?

Explore the distinction between the expressions (-x)^y and -x^y. This problem focuses on exponentiation with negative bases and demonstrates how the placement of the negative sign impacts the result. Suitable for students in Grades 8-10.

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