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
Here are the solutions to the problems. i) Given sqrt(3)+sqrt(5)sqrt(5) = (x+y)sqrt(15), find (x+y). Step 1: Simplify the left-hand side (LHS) of the equation. sqrt(3)+sqrt(5)sqrt(5) = sqrt(3)sqrt(5) + sqrt(5)sqrt(5) = sqrt(3)sqrt(5) + 1 Step 2: Rationalize the denominator of the first term in the LHS. sqrt(3)sqrt(5) × sqrt(5)sqrt(5) = sqrt(3 × 5)5 = sqrt(15)5 So, the LHS becomes: sqrt(15)5 + 1 Step 3: Equate the simplified LHS to the right-hand side (RHS) and solve for (x+y). sqrt(15)5 + 1 = (x+y)sqrt(15) To isolate (x+y), divide both sides by sqrt(15): (x+y) = sqrt(15)5 + 1sqrt(15) Combine the terms in the numerator: (x+y) = sqrt(15) + 55sqrt(15) = sqrt(15) + 55sqrt(15) Step 4: Rationalize the denominator for the expression of (x+y). Multiply the numerator and denominator by sqrt(15): (x+y) = sqrt(15) + 55sqrt(15) × sqrt(15)sqrt(15) (x+y) = (sqrt(15))(sqrt(15)) + 5sqrt(15)5(sqrt(15))(sqrt(15)) (x+y) = 15 + 5sqrt(15)5 × 15 (x+y) = 15 + 5sqrt(15)75 Step 5: Simplify the expression by dividing each term in the numerator by the denominator. (x+y) = (15)/(75) + 5sqrt(15)75 (x+y) = (1)/(5) + sqrt(15)15 The value of (x+y) is: (1)/(5) + sqrt(15)15 ii) Find sqrt(15)+55 - sqrt(15)+55 Step 1: Identify the terms in the expression. The expression consists of two identical terms being subtracted from each other. Let A = sqrt(15)+55. The expression is A - A. Step 2: Perform the subtraction. When an expression is subtracted from itself, the result is zero. sqrt(15)+55 - sqrt(15)+55 = 0 The simplified expression is: 0 3 done, 2 left today. You're making progress.