This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To express (2x - 3y)^4, we use the binomial theorem, which states that (a+b)^n = _k=0^n nk a^n-k b^k. Here, a = 2x, b = -3y, and n = 4. Step 1: Write out the terms using the binomial theorem. The expansion will have n+1 = 5 terms: 40(2x)^4(-3y)^0 + 41(2x)^3(-3y)^1 + 42(2x)^2(-3y)^2 + 43(2x)^1(-3y)^3 + 44(2x)^0(-3y)^4 Step 2: Calculate the binomial coefficients nk. 40 = 1 41 = 4 42 = (4 × 3)/(2 × 1) = 6 43 = 4 44 = 1 Step 3: Expand each term. For k=0: 1 · (2x)^4 · (-3y)^0 = 1 · (16x^4) · 1 = 16x^4 For k=1: 4 · (2x)^3 · (-3y)^1 = 4 · (8x^3) · (-3y) = -96x^3y For k=2: 6 · (2x)^2 · (-3y)^2 = 6 · (4x^2) · (9y^2) = 216x^2y^2 For k=3: 4 · (2x)^1 · (-3y)^3 = 4 · (2x) · (-27y^3) = -216xy^3 For k=4: 1 · (2x)^0 · (-3y)^4 = 1 · 1 · (81y^4) = 81y^4 Step 4: Combine all the terms. (2x - 3y)^4 = 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4 The expanded expression is: 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4