Which of the following equations is true for all nonzero numbers m and n?

This equation is incorrect as it misrepresents the properties of exponents. The correct exponential property states that \(3^{m+n} = 3^m \cdot 3^n\), not a sum. Therefore, this equation does not hold for any nonzero values of m and n.

This equation is also false due to the misapplication of the square of a sum. The correct expansion is \((m+n)^2 = m^2 + 2mn + n^2\), which includes an additional term that is essential for accuracy. Thus, this equality does not hold for all nonzero m and n.

This equation is incorrect because it neglects the properties of square roots and sums of squares. The correct relationship is given by the triangle inequality, stating that \(\sqrt{m^2 + n^2} \leq m + n\), which cannot be considered equal in general. Therefore, this equality is not valid for all nonzero numbers.

This equation is incorrect; it misrepresents the process of adding fractions. The left-hand side represents a single fraction, while the right-hand side is the sum of two separate fractions. The correct form involves a common denominator, thus making this equation false for nonzero m and n.