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 find the term independent of p in the binomial expansion of ((3p^2)/(2) - (1)/(3p))^9, we use the general term formula for a binomial expansion (a+b)^n, which is T_r+1 = nr a^n-r b^r. Step 1: Identify a, b, and n. In this expansion, a = (3p^2)/(2), b = -(1)/(3p), and n = 9. Step 2: Write the general term T_r+1. T_r+1 = 9r ((3p^2)/(2))^9-r (-(1)/(3p))^r Step 3: Separate the coefficients and the powers of p. T_r+1 = 9r ((3)/(2))^9-r (p^2)^9-r (-(1)/(3))^r (p^-1)^r T_r+1 = 9r ((3)/(2))^9-r (-(1)/(3))^r p^2(9-r) p^-r T_r+1 = 9r ((3)/(2))^9-r (-(1)/(3))^r p^18-2r-r T_r+1 = 9r ((3)/(2))^9-r (-(1)/(3))^r p^18-3r Step 4: Find the value of r for the term independent of p. For the term to be independent of p, the exponent of p must be 0. 18 - 3r = 0 3r = 18 r = 6 Step 5: Substitute r=6 back into the general term to find the constant term. The term independent of p is T_6+1 = T_7. T_7 = 96 ((3)/(2))^9-6 (-(1)/(3))^6 T_7 = 96 ((3)/(2))^3 (-(1)/(3))^6 Step 6: Calculate the binomial coefficient and the powers. 96 = 93 = (9 × 8 × 7)/(3 × 2 × 1) = 3 × 4 × 7 = 84 ((3)/(2))^3 = (3^3)/(2^3) = (27)/(8) (-(1)/(3))^6 = ((-1)^6)/(3^6) = (1)/(729) Step 7: Multiply the calculated values. T_7 = 84 × (27)/(8) × (1)/(729) T_7 = (84 × 27)/(8 × 729) Simplify the expression: T_7 = (84)/(8) × (27)/(729) T_7 = (21)/(2) × (1)/(27) T_7 = (21)/(54) Divide both numerator and denominator by their greatest common divisor, which is 3: T_7 = (21 ÷ 3)/(54 ÷ 3) = (7)/(18) Step 8: Compare the result with the given options. The calculated term is (7)/(18), which corresponds to option D. The final answer is D. Last free one today — make it count tomorrow, or type /upgrade for unlimited.