g) \( (3 x+2 y)^{3} \)

To expand \( (3x + 2y)^{3} \), you can use the binomial theorem, which states that \( (a + b)^{n} = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \). Here, \( a = 3x \), \( b = 2y \), and \( n = 3 \). Expanding this gives you: \[ (3x + 2y)^{3} = \binom{3}{0}(3x)^{3}(2y)^{0} + \binom{3}{1}(3x)^{2}(2y)^{1} + \binom{3}{2}(3x)^{1}(2y)^{2} + \binom{3}{3}(3x)^{0}(2y)^{3} \] Calculating each term, you get: \[ 27x^{3} + 54x^{2}y + 36xy^{2} + 8y^{3} \] Thus, the final expansion is: \[ (3x + 2y)^{3} = 27x^{3} + 54x^{2}y + 36xy^{2} + 8y^{3} \] Dive into the world of polynomials! Expanding expressions like this not only strengthens your algebra skills, but it's a key step in calculus when dealing with derivatives and integrals. Plus, it turns these expressions into delightful combinations that can make various math challenges much easier! If you’re interested in more complex expansions, consider diving into multinomial expansions, where you can tackle equations with more than two terms. It’s like leveling up your polynomial skills! For extra fun, try visualizing the expansion process with a simple programming language or even by hand to see how the coefficients and terms unfold.