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
Step 1: Identify the expression to rationalize. The expression is (2)/(3+sqrt(5)). Step 2: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 3+sqrt(5) is 3-sqrt(5). (2)/(3+sqrt(5)) × 3-sqrt(5)3-sqrt(5) Step 3: Expand the numerator. 2(3-sqrt(5)) = 6 - 2sqrt(5) Step 4: Expand the denominator using the difference of squares formula (a+b)(a-b) = a^2 - b^2. (3+sqrt(5))(3-sqrt(5)) = 3^2 - (sqrt(5))^2 = 9 - 5 = 4 Step 5: Combine the simplified numerator and denominator. 6 - 2sqrt(5)4 Step 6: Simplify the fraction by dividing each term in the numerator by the denominator. (6)/(4) - 2sqrt(5)4 = (3)/(2) - sqrt(5)2 This can also be written as: 3-sqrt(5)2 The rationalized and simplified expression is 3-sqrt(5)2. 3 done, 2 left today. You're making progress.