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: Rationalize the denominator. To rationalize the denominator sqrt(5) - sqrt(2), multiply the numerator and denominator by its conjugate, which is sqrt(5) + sqrt(2). sqrt(2) + 2sqrt(5)sqrt(5) - sqrt(2) × sqrt(5) + sqrt(2)sqrt(5) + sqrt(2) Step 2: Expand the numerator. (sqrt(2) + 2sqrt(5))(sqrt(5) + sqrt(2)) = sqrt(2)sqrt(5) + sqrt(2)sqrt(2) + 2sqrt(5)sqrt(5) + 2sqrt(5)sqrt(2) = sqrt(10) + 2 + 2(5) + 2sqrt(10) = sqrt(10) + 2 + 10 + 2sqrt(10) = 12 + 3sqrt(10) Step 3: Expand the denominator. Use the difference of squares formula (a-b)(a+b) = a^2 - b^2: (sqrt(5) - sqrt(2))(sqrt(5) + sqrt(2)) = (sqrt(5))^2 - (sqrt(2))^2 = 5 - 2 = 3 Step 4: Combine the simplified numerator and denominator. 12 + 3sqrt(10)3 Step 5: Simplify the expression by dividing by the common factor 3. (12)/(3) + 3sqrt(10)3 = 4 + sqrt(10) The simplified expression is 4 + sqrt(10). What's next?