This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
ScanSolve AI Answer
You are correct that the combination method nk is used to find the coefficients, which are the same numbers found in Pascal's Triangle. I will complete the expansion of (3+5x)^6 using the coefficients from Pascal's Triangle. Step 1: Identify the coefficients from Pascal's Triangle for n=6. The 6th row of Pascal's Triangle (starting with row 0) provides the coefficients: 1, 6, 15, 20, 15, 6, 1 Step 2: Apply these coefficients to the terms of the binomial expansion (a+b)^n, where a=3 and b=5x. The general form is C_k a^n-k b^k, where C_k is the coefficient from Pascal's Triangle. Term 1 (k=0): Coefficient is 1. 1 · (3)^6-0 · (5x)^0 = 1 · 3^6 · 1 = 1 · 729 · 1 = 729 Term 2 (k=1): Coefficient is 6. 6 · (3)^6-1 · (5x)^1 = 6 · 3^5 · 5x = 6 · 243 · 5x = 6 · 1215x = 7290x Term 3 (k=2): Coefficient is 15. 15 · (3)^6-2 · (5x)^2 = 15 · 3^4 · (25x^2) = 15 · 81 · 25x^2 = 15 · 2025x^2 = 30375x^2 Term 4 (k=3): Coefficient is 20. 20 · (3)^6-3 · (5x)^3 = 20 · 3^3 · (125x^3) = 20 · 27 · 125x^3 = 20 · 3375x^3 = 67500x^3 Term 5 (k=4): Coefficient is 15. 15 · (3)^6-4 · (5x)^4 = 15 · 3^2 · (625x^4) = 15 · 9 · 625x^4 = 15 · 5625x^4 = 84375x^4 Term 6 (k=5): Coefficient is 6. 6 · (3)^6-5 · (5x)^5 = 6 · 3^1 · (3125x^5) = 6 · 3 · 3125x^5 = 18 · 3125x^5 = 56250x^5 Term 7 (k=6): Coefficient is 1. 1 · (3)^6-6 · (5x)^6 = 1 · 3^0 · (15625x^6) = 1 · 1 · 15625x^6 = 15625x^6 Step 3: Sum all the terms to get the full expansion. (3+5x)^6 = 729 + 7290x + 30375x^2 + 67500x^3 + 84375x^4 + 56250x^5 + 15625x^6 The final answer is 729 + 7290x + 30375x^2 + 67500x^3 + 84375x^4 + 56250x^5 + 15625x^6.