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
Here are the expanded forms of the given expressions. i) We need to expand ((3)/(2)x + 1)^3. We use the identity (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. Here, a = (3)/(2)x and b = 1. Step 1: Substitute a and b into the formula. ((3)/(2)x + 1)^3 = ((3)/(2)x)^3 + 3((3)/(2)x)^2(1) + 3((3)/(2)x)(1)^2 + (1)^3 Step 2: Simplify each term. = (3^3)/(2^3)x^3 + 3((3^2)/(2^2)x^2)(1) + 3((3)/(2)x)(1) + 1 = (27)/(8)x^3 + 3((9)/(4)x^2) + (9)/(2)x + 1 Step 3: Perform the multiplications. = (27)/(8)x^3 + (27)/(4)x^2 + (9)/(2)x + 1 The expanded form is: (27)/(8)x^3 + (27)/(4)x^2 + (9)/(2)x + 1 ii) We need to expand ((x)/(2) - (y)/(3))^3. We use the identity (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. Here, a = (x)/(2) and b = (y)/(3). Step 1: Substitute a and b into the formula. ((x)/(2) - (y)/(3))^3 = ((x)/(2))^3 - 3((x)/(2))^2((y)/(3)) + 3((x)/(2))((y)/(3))^2 - ((y)/(3))^3 Step 2: Simplify each term. = (x^3)/(2^3) - 3((x^2)/(2^2))((y)/(3)) + 3((x)/(2))((y^2)/(3^2)) - (y^3)/(3^3) = (x^3)/(8) - 3((x^2)/(4))((y)/(3)) + 3((x)/(2))((y^2)/(9)) - (y^3)/(27) Step 3: Perform the multiplications and simplify the fractions. = (x^3)/(8) - (3x^2y)/(12) + (3xy^2)/(18) - (y^3)/(27) = (x^3)/(8) - (x^2y)/(4) + (xy^2)/(6) - (y^3)/(27) The expanded form is: (x^3)/(8) - (x^2y)/(4) + (xy^2)/(6) - (y^3)/(27) That's 2 down. 3 left today — send the next one.