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
Hey R.LAMAR😇😈, good to see you again. Here's the solution to your question: The binary operation is defined as p * q = (3p + 2q)/(2p + 3q). a) Evaluate (sqrt(3)) * (sqrt(2)) Step 1: Substitute p = sqrt(3) and q = sqrt(2) into the given formula. (sqrt(3)) * (sqrt(2)) = 3(sqrt(3)) + 2(sqrt(2))2(sqrt(3)) + 3(sqrt(2)) Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2sqrt(3) - 3sqrt(2). (sqrt(3)) * (sqrt(2)) &= 3sqrt(3) + 2sqrt(2)2sqrt(3) + 3sqrt(2) × 2sqrt(3) - 3sqrt(2)2sqrt(3) - 3sqrt(2) \\ &= (3sqrt(3))(2sqrt(3)) - (3sqrt(3))(3sqrt(2)) + (2sqrt(2))(2sqrt(3)) - (2sqrt(2))(3sqrt(2))(2sqrt(3))^2 - (3sqrt(2))^2 \\ &= 6(3) - 9sqrt(6) + 4sqrt(6) - 6(2)4(3) - 9(2) \\ &= 18 - 9sqrt(6) + 4sqrt(6) - 1212 - 18 \\ &= 6 - 5sqrt(6)-6 Step 3: Simplify the expression. (sqrt(3)) * (sqrt(2)) = -(5sqrt(6) - 6)-6 = 5sqrt(6) - 66 The value is 5sqrt(6) - 66. b) Find the value of x for which x * 3 = (1)/(2) Step 1: Substitute p = x and q = 3 into the given formula. x * 3 = (3(x) + 2(3))/(2(x) + 3(3)) = (3x + 6)/(2x + 9) Step 2: Set the expression equal to (1)/(2) and solve for x. (3x + 6)/(2x + 9) = (1)/(2) Step 3: Cross-multiply to eliminate the denominators. 2(3x + 6) = 1(2x + 9) 6x + 12 = 2x + 9 Step 4: Collect like terms and solve for x. 6x - 2x = 9 - 12 4x = -3 x = -(3)/(4) The value of x is -(3)/(4). Send me the next one 📸