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: Recall the values of 30^ and 60^. 30^ = sqrt(3)2 60^ = sqrt(3)2 Step 2: Substitute these values into the given expression. (1 + 30^)/(1 - 60^) = 1 + sqrt(3)21 - sqrt(3)2 Step 3: Simplify the numerator and the denominator by finding a common denominator. (2)/(2) + sqrt(3)2(2)/(2) - sqrt(3)2 = 2 + sqrt(3)22 - sqrt(3)2 Step 4: Simplify the complex fraction. 2 + sqrt(3)2 - sqrt(3) Step 5: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2 + sqrt(3). 2 + sqrt(3)2 - sqrt(3) × 2 + sqrt(3)2 + sqrt(3) Step 6: Expand the numerator and the denominator. Numerator: (2 + sqrt(3))(2 + sqrt(3)) = 2^2 + 2sqrt(3) + 2sqrt(3) + (sqrt(3))^2 = 4 + 4sqrt(3) + 3 = 7 + 4sqrt(3) Denominator: (2 - sqrt(3))(2 + sqrt(3)) = 2^2 - (sqrt(3))^2 = 4 - 3 = 1 Step 7: Write the final simplified expression. 7 + 4sqrt(3)1 = 7 + 4sqrt(3) The rationalized expression is 7 + 4sqrt(3). Send me the next one 📸