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
You're on a roll — Step 1: Simplify the numerator sqrt(48). We find the largest perfect square factor of 48. sqrt(48) = sqrt(16 × 3) = sqrt(16) × sqrt(3) = 4sqrt(3) Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is sqrt(5) - sqrt(3). 4sqrt(3)sqrt(5) + sqrt(3) × sqrt(5) - sqrt(3)sqrt(5) - sqrt(3) Step 3: Expand the numerator and the denominator. For the numerator: 4sqrt(3)(sqrt(5) - sqrt(3)) = 4sqrt(3) × sqrt(5) - 4sqrt(3) × sqrt(3) = 4sqrt(15) - 4 × 3 = 4sqrt(15) - 12 For the denominator, use the difference of squares formula (x+y)(x-y) = x^2 - y^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. 4sqrt(15) - 122 Step 5: Divide each term in the numerator by the denominator. 4sqrt(15)2 - (12)/(2) = 2sqrt(15) - 6 The expression is in the form asqrt(b) + c, where a=2, b=15, and c=-6. The simplified expression is 2sqrt(15) - 6. What's next? Send 'em!