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
Okay, let's do a binomial expansion problem. Expand (2x - 3y)^4. Step 1: Recall the binomial theorem formula for (a+b)^n: (a+b)^n = _k=0^n nk a^n-k b^k In this case, a = 2x, b = -3y, and n = 4. Step 2: Apply the formula for each term from k=0 to k=4. For k=0: 40 (2x)^4-0 (-3y)^0 = 1 · (2x)^4 · 1 = 16x^4 For k=1: 41 (2x)^4-1 (-3y)^1 = 4 · (2x)^3 · (-3y) = 4 · 8x^3 · (-3y) = -96x^3y For k=2: 42 (2x)^4-2 (-3y)^2 = 6 · (2x)^2 · (9y^2) = 6 · 4x^2 · 9y^2 = 216x^2y^2 For k=3: 43 (2x)^4-3 (-3y)^3 = 4 · (2x)^1 · (-27y^3) = 4 · 2x · (-27y^3) = -216xy^3 For k=4: 44 (2x)^4-4 (-3y)^4 = 1 · (2x)^0 · (81y^4) = 1 · 1 · 81y^4 = 81y^4 Step 3: Combine all the terms. (2x - 3y)^4 = 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4 The final answer is 16x^4 - 96x^3y + 216x^2y^2 - 216xy^3 + 81y^4. That's 2 down. 3 left today — send the next one.