If a, be R, show that a ^ 2 + b ^ 2 = 0 if and only if a = 0 and b = 0.

We are asked to show that \( a^2 + b^2 = 0 \) if and only if \( a = 0 \) and \( b = 0 \), where \( a, b \in \mathbb{R} \).

### Part 1: Show that \( a^2 + b^2 = 0 \) implies \( a = 0 \) and \( b = 0 \)

- Since \( a^2 \geq 0 \) and \( b^2 \geq 0 \) for any real numbers \( a \) and \( b \), their sum \( a^2 + b^2 \geq 0 \).
- The only way \( a^2 + b^2 = 0 \) can hold is if both \( a^2 = 0 \) and \( b^2 = 0 \) because the sum of two non-negative numbers can only be zero if both numbers are zero.
- Therefore, \( a = 0 \) and \( b = 0 \).

### Part 2: Show that \( a = 0 \) and \( b = 0 \) implies \( a^2 + b^2 = 0 \)

- If \( a = 0 \) and \( b = 0 \), then \( a^2 = 0^2 = 0 \) and \( b^2 = 0^2 = 0 \).
- Therefore, \( a^2 + b^2 = 0 + 0 = 0 \).

### Conclusion:

We have shown both directions of the statement:
- If \( a^2 + b^2 = 0 \), then \( a = 0 \) and \( b = 0 \).
- If \( a = 0 \) and \( b = 0 \), then \( a^2 + b^2 = 0 \).

Thus, \( a^2 + b^2 = 0 \) if and only if \( a = 0 \) and \( b = 0 \).

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Do you need further details on this? 

Here are 5 related questions you might want to explore:
1. Can the result change if \( a \) and \( b \) are complex numbers?
2. What would \( a^2 + b^2 = 0 \) imply if \( a, b \in \mathbb{C} \)?
3. How does this result extend to three dimensions, i.e., \( a^2 + b^2 + c^2 = 0 \)?
4. What happens if we work with vectors instead of scalars for \( a \) and \( b \)?
5. Can this be generalized to norms in higher-dimensional spaces?

**Tip:** When dealing with sums of squares, always remember that for real numbers, the square of any number is non-negative, and equality to zero forces each term to be zero.

If a, b ∈ R, show that a^2 + b^2 = 0 if and only if a = 0 and b = 0.

Explore the proof that a^2 + b^2 = 0 implies a = 0 and b = 0, focusing on real numbers and the non-negativity of squares. This detailed solution covers key mathematical concepts such as inequalities and implication proofs, suitable for high school students (Grades 9-12).

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