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
The expression given is (3 × 5x)^6. Pascal's method (or Pascal's Triangle) is used for expanding binomials (expressions with two terms, like (a+b)^n). Since (3 × 5x)^6 simplifies to a monomial (15x)^6, Pascal's method is not directly applicable to the expression as written. Here is the direct solution for the given expression: Step 1: Simplify the expression inside the parenthesis. (3 × 5x)^6 = (15x)^6 Step 2: Apply the exponent to both factors inside the parenthesis. (15x)^6 = 15^6 × x^6 Step 3: Calculate 15^6. 15^6 = 15 × 15 × 15 × 15 × 15 × 15 = 11,390,625 Step 4: Combine the results. (15x)^6 = 11,390,625x^6 The final answer is 11,390,625x^6. --- Assuming the problem intended to be (3+5x)^6 (a binomial expansion): If the problem intended to be (3+5x)^6, then Pascal's method would be used to find the coefficients for the binomial expansion. 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 the binomial theorem using a=3 and b=5x. The general form for (a+b)^n is _k=0^n nk a^n-k b^k. Term 1 (k=0): 60 (3)^6 (5x)^0 = 1 · (729) · (1) = 729 Term 2 (k=1): 61 (3)^5 (5x)^1 = 6 · (243) · (5x) = 6 · 1215x = 7290x Term 3 (k=2): 62 (3)^4 (5x)^2 = 15 · (81) · (25x^2) = 15 · 2025x^2 = 30375x^2