A teacher showed 5 expressions to her students. (a+b)^2, (a-b)^2, (b-a)^2, (-a+b)^2, and (-a-b)^2. Without finding the expansion, determine which, if any, of the expressions are equal. Show your working clearly.

To determine which of the expressions are equal without directly expanding them, we can analyze the relationships between the variables \( a \) and \( b \) in each expression. 

### Expressions Given
1. \( (a + b)^2 \)
2. \( (a - b)^2 \)
3. \( (b - a)^2 \)
4. \( (-a + b)^2 \)
5. \( (-a - b)^2 \)

### Step 1: Identify the equivalent expressions by examining the forms

1. **Expression 1:** \( (a + b)^2 \)

2. **Expression 2:** \( (a - b)^2 \)

3. **Expression 3:** \( (b - a)^2 \)

   Notice that \( (b - a)^2 \) can be rewritten as \( (-(a - b))^2 \). Since squaring a negative number gives the same result as squaring the positive version, we have:
   \[
   (b - a)^2 = (-(a - b))^2 = (a - b)^2
   \]
   Therefore, \( (b - a)^2 = (a - b)^2 \).

4. **Expression 4:** \( (-a + b)^2 \)

   This can be rewritten as \( (b - a)^2 \), which, as we established, is equal to \( (a - b)^2 \).

5. **Expression 5:** \( (-a - b)^2 \)

   This can be rewritten as \( (-(a + b))^2 \). Again, squaring the negative of a term gives the same result as squaring the positive term:
   \[
   (-a - b)^2 = (a + b)^2
   \]

### Step 2: Group the equivalent expressions

Based on the above analysis:
- **\( (a + b)^2 \)** and **\( (-a - b)^2 \)** are equal.
- **\( (a - b)^2 \)**, **\( (b - a)^2 \)**, and **\( (-a + b)^2 \)** are equal.

### Conclusion
- The expressions **\( (a + b)^2 \)** and **\( (-a - b)^2 \)** are equal.
- The expressions **\( (a - b)^2 \)**, **\( (b - a)^2 \)**, and **\( (-a + b)^2 \)** are equal.

Would you like further clarification on this analysis or have any other questions?

Here are 5 related questions you might find helpful:
1. Why does squaring a negative term result in the same value as squaring the positive term?
2. How do you determine if two algebraic expressions are equal without expanding them?
3. What is the importance of the commutative property in identifying equal expressions?
4. How does changing the sign inside a binomial affect its square?
5. What are some other examples of equivalent expressions involving squares?

**Tip:** When comparing expressions involving squares, remember that the order of terms inside a binomial does not affect the square, and changing the sign inside a binomial does not change the result when squared.

Learn how to determine the equality of algebraic expressions without expanding them directly. This problem involves analyzing the relationships between variables in squared expressions. Suitable for high school students studying algebra.

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