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
Hey, good to see you again. Step 1: Recall the algebraic identity for the cube of a sum. The identity is (a+b)^3 = a^3 + b^3 + 3ab(a+b). Step 2: Apply the identity to the given expression. Let a = x and b = (1)/(x). Then, a+b = x + (1)/(x) and ab = x · (1)/(x) = 1. Substitute these into the identity: (x + (1)/(x))^3 = x^3 + ((1)/(x))^3 + 3 · x · (1)/(x) (x + (1)/(x)) (x + (1)/(x))^3 = x^3 + (1)/(x^3) + 3 (x + (1)/(x)) Step 3: Substitute the given value into the equation. We are given that x + (1)/(x) = 3. Substitute this value into the equation from Step 2: (3)^3 = x^3 + (1)/(x^3) + 3(3) 27 = x^3 + (1)/(x^3) + 9 Step 4: Solve for x^3 + (1)/(x^3). Subtract 9 from both sides of the equation: x^3 + (1)/(x^3) = 27 - 9 x^3 + (1)/(x^3) = 18 The value of x^3 + (1)/(x^3) is: 18 Send me the next one 📸