Question 16

(m + n)/3 = m/3 + n/3

The equation (m + n)/3 = m/3 + n/3 is true for all nonzero numbers m and n, as it illustrates the distributive property of division over addition.

A) 3^(m+n) = 3^m + 3^n

This equation is incorrect because it does not hold true for all nonzero numbers m and n. The correct property for exponents states that 3^(m+n) = 3^m * 3^n, not the sum of the two exponential terms.

B) (m + n)^2 = m^2 + n^2

This equation is also false for all nonzero m and n. The correct expansion of (m + n)^2 is m^2 + 2mn + n^2, which includes the cross-product term 2mn, showing that the original equation does not hold.

C) sqrt(m^2 + n^2) = m + n

This statement is incorrect as well. The square root of the sum of squares, sqrt(m^2 + n^2), is typically less than or equal to m + n due to the triangle inequality, particularly when both m and n are positive.

D) 3/(m + n) = 3/m + 3/n

This equation is false as it misapplies the properties of fractions. The correct relationship would involve a common denominator, yielding 3/(m+n) = 3(mn)/(m+n), which does not equate to 3/m + 3/n.

E) (m + n)/3 = m/3 + n/3

This equation is true by the distributive property, showing that dividing the sum of m and n by 3 is equivalent to dividing each term individually by 3, hence it holds for all nonzero values of m and n.

Conclusion

The equation (m + n)/3 = m/3 + n/3 is definitively correct as it adheres to the distributive property of division. All other options fail to hold true universally for nonzero m and n due to incorrect applications of algebraic properties or fundamental mathematical principles. Thus, E is the only valid choice.