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: Find a common denominator for the two fractions. The denominators are uv and v^2. The least common multiple (LCM) of uv and v^2 is uv^2. Step 2: Rewrite each fraction with the common denominator. For the first fraction, multiply the numerator and denominator by v: (u^2 - v^2)/(uv) = ((u^2 - v^2) · v)/(uv · v) = (u^2v - v^3)/(uv^2) For the second fraction, multiply the numerator and denominator by u: (v - 3uv - u^2)/(v^2) = ((v - 3uv - u^2) · u)/(v^2 · u) = (uv - 3u^2v - u^3)/(uv^2) Step 3: Add the rewritten fractions. (u^2v - v^3)/(uv^2) + (uv - 3u^2v - u^3)/(uv^2) = (u^2v - v^3 + uv - 3u^2v - u^3)/(uv^2) Step 4: Combine like terms in the numerator. Combine the terms involving u^2v: u^2v - 3u^2v = -2u^2v. The numerator becomes: -u^3 - 2u^2v + uv - v^3. Step 5: Write the simplified expression. (-u^3 - 2u^2v + uv - v^3)/(uv^2) The final answer is (-u^3 - 2u^2v + uv - v^3)/(uv^2).