Show that the statement 'For any real numbers a and b, `a^(2)=b^(2)` implies that a = b is not true' by giving counter example.

Answer

To show that the statement "For any real numbers \( a \) and \( b \), \( a^2 = b^2 \) implies that \( a = b \)" is not true, we can provide a counterexample.

### Step-by-Step Solution:

1. **Understanding the Statement**: The statement claims that if the squares of two real numbers are equal, then the numbers themselves must be equal. We need to find a case where this is not true.

2. **Choosing Values for \( a \) and \( b \)**: Let's choose \( a = 2 \) and \( b = -2 \). These are two different real numbers.

3. **Calculating the Squares**:
- Calculate \( a^2 \):
\[ a^2 = 2^2 = 4 \]
- Calculate \( b^2 \):
\[ b^2 = (-2)^2 = 4 \]

4. **Comparing the Squares**: Now we see that:
\[ a^2 = b^2 \implies 4 = 4 \]
This shows that \( a^2 = b^2 \) holds true.

5. **Checking the Equality of \( a \) and \( b \)**: Now, let's check if \( a = b \):
\[ a = 2 \quad \text{and} \quad b = -2 \]
Clearly, \( 2 \neq -2 \). Thus, \( a \neq b \).

6. **Conclusion**: Since we have found \( a \) and \( b \) such that \( a^2 = b^2 \) but \( a \neq b \), we conclude that the statement "For any real numbers \( a \) and \( b \), \( a^2 = b^2 \) implies that \( a = b \)" is not true.

### Final Statement:
Therefore, the counterexample \( a = 2 \) and \( b = -2 \) demonstrates that \( a^2 = b^2 \) does not imply \( a = b \).

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