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: Square the given equation to find x^2 + (1)/(x^2). Given the equation: x + (1)/(x) = sqrt(3) Square both sides of the equation: (x + (1)/(x))^2 = (sqrt(3))^2 Expand the left side using the formula (a+b)^2 = a^2 + 2ab + b^2: x^2 + 2 · x · (1)/(x) + ((1)/(x))^2 = 3 x^2 + 2 + (1)/(x^2) = 3 Subtract 2 from both sides: x^2 + (1)/(x^2) = 3 - 2 x^2 + (1)/(x^2) = 1 Step 2: Square the result from Step 1 to find x^4 + (1)/(x^4). We have x^2 + (1)/(x^2) = 1. Square both sides of this equation: (x^2 + (1)/(x^2))^2 = (1)^2 Expand the left side: (x^2)^2 + 2 · x^2 · (1)/(x^2) + ((1)/(x^2))^2 = 1 x^4 + 2 + (1)/(x^4) = 1 Subtract 2 from both sides: x^4 + (1)/(x^4) = 1 - 2 x^4 + (1)/(x^4) = -1 Step 3: Rewrite the expression to be evaluated. The expression to evaluate is (x^8 + 1)/(x^4). We can split this fraction: (x^8 + 1)/(x^4) = (x^8)/(x^4) + (1)/(x^4) Simplify the first term: (x^8 + 1)/(x^4) = x^8-4 + (1)/(x^4) (x^8 + 1)/(x^4) = x^4 + (1)/(x^4) Step 4: Substitute the value found in Step 2. From Step 2, we found that x^4 + (1)/(x^4) = -1. Therefore, (x^8 + 1)/(x^4) = -1 The final answer is -1. Send me the next one 📸