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 rationalize the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The denominator is 4sqrt(3) - 2sqrt(2), so its conjugate is 4sqrt(3) + 2sqrt(2). Step 1: Multiply the expression by the conjugate. 3sqrt(2) - 4sqrt(3)4sqrt(3) - 2sqrt(2) × 4sqrt(3) + 2sqrt(2)4sqrt(3) + 2sqrt(2) Step 2: Expand the numerator. (3sqrt(2) - 4sqrt(3))(4sqrt(3) + 2sqrt(2)) (3sqrt(2))(4sqrt(3)) + (3sqrt(2))(2sqrt(2)) - (4sqrt(3))(4sqrt(3)) - (4sqrt(3))(2sqrt(2)) 12sqrt(6) + 6(sqrt(2))^2 - 16(sqrt(3))^2 - 8sqrt(6) 12sqrt(6) + 6(2) - 16(3) - 8sqrt(6) 12sqrt(6) + 12 - 48 - 8sqrt(6) Combine like terms: (12 - 48) + (12sqrt(6) - 8sqrt(6)) -36 + 4sqrt(6) Step 3: Expand the denominator. This is in the form (a-b)(a+b) = a^2 - b^2. (4sqrt(3) - 2sqrt(2))(4sqrt(3) + 2sqrt(2)) = (4sqrt(3))^2 - (2sqrt(2))^2 (16 × 3) - (4 × 2) 48 - 8 40 Step 4: Combine the simplified numerator and denominator. -36 + 4sqrt(6)40 Step 5: Simplify the fraction by dividing each term in the numerator by the denominator. Divide both terms by their greatest common factor, which is 4. (-36)/(40) + 4sqrt(6)40 -(9)/(10) + sqrt(6)10 This can also be written as: sqrt(6) - 910 That's 2 down. 3 left today — send the next one.