Write the binomial expansion of the expression. \( (4 m+3 n)^{4} \)

The binomial expansion of \( (4m + 3n)^{4} \) can be found using the binomial theorem, which tells us that \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \). For this expression, \( a = 4m \), \( b = 3n \), and \( n = 4 \). Expanding it gives: \[ (4m + 3n)^{4} = \binom{4}{0} (4m)^{4} (3n)^{0} + \binom{4}{1} (4m)^{3} (3n)^{1} + \binom{4}{2} (4m)^{2} (3n)^{2} + \binom{4}{3} (4m)^{1} (3n)^{3} + \binom{4}{4} (4m)^{0} (3n)^{4} \] Calculating each term yields: \[ = 1 \cdot (256m^4) + 4 \cdot (48m^3 n) + 6 \cdot (36m^2 n^2) + 4 \cdot (12m n^3) + 1 \cdot (81n^4) \] So, simplifying this results in: \[ = 256m^4 + 192m^3 n + 216m^2 n^2 + 48m n^3 + 81n^4 \] Thus, the binomial expansion is: \[ 256m^4 + 192m^3 n + 216m^2 n^2 + 48m n^3 + 81n^4 \]