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 conjugate of sqrt(5) - sqrt(3) is sqrt(5) + sqrt(3). Step 1: Multiply by the conjugate. 6sqrt(2) - 2sqrt(10)sqrt(5) - sqrt(3) × sqrt(5) + sqrt(3)sqrt(5) + sqrt(3) Step 2: Expand the numerator. (6sqrt(2) - 2sqrt(10))(sqrt(5) + sqrt(3)) (6sqrt(2))(sqrt(5)) + (6sqrt(2))(sqrt(3)) - (2sqrt(10))(sqrt(5)) - (2sqrt(10))(sqrt(3)) 6sqrt(10) + 6sqrt(6) - 2sqrt(50) - 2sqrt(30) Simplify sqrt(50): sqrt(50) = sqrt(25 × 2) = 5sqrt(2). Substitute this back into the expression: 6sqrt(10) + 6sqrt(6) - 2(5sqrt(2)) - 2sqrt(30) 6sqrt(10) + 6sqrt(6) - 10sqrt(2) - 2sqrt(30) Step 3: Expand the denominator. This is in the form (a-b)(a+b) = a^2 - b^2. (sqrt(5) - sqrt(3))(sqrt(5) + sqrt(3)) = (sqrt(5))^2 - (sqrt(3))^2 5 - 3 2 Step 4: Combine the simplified numerator and denominator. 6sqrt(10) + 6sqrt(6) - 10sqrt(2) - 2sqrt(30)2 Step 5: Simplify the fraction by dividing each term in the numerator by the denominator. 6sqrt(10)2 + 6sqrt(6)2 - 10sqrt(2)2 - 2sqrt(30)2 3sqrt(10) + 3sqrt(6) - 5sqrt(2) - sqrt(30) The rationalized expression is: 3sqrt(10) + 3sqrt(6) - 5sqrt(2) - sqrt(30) 3 done, 2 left today. You're making progress.