\frac{1}{\sqrt{n} + \sqrt{n+1}} \cdot \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} = \frac{\sqrt{n+1} - \sqrt{n}}{(n+1) - n} = \sqrt{n+1} - \sqrt{n}

["# Simplify the Expression: How to Evaluate\n$$ \frac{1}{\sqrt{n} + \sqrt{n+1}} \cdot \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} $$", "Mathematical expressions often appear complex at first glance, but with careful algebraic manipulation—especially rationalizing denominators—a nice simplification emerges. In this article, we’ll break down the expression\n$$ \frac{1}{\sqrt{n} + \sqrt{n+1}} \cdot \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} $$\nand demonstrate how it simplifies to\n$$ \sqrt{n+1} - \sqrt{n}. $$", "---", "## Step 1: Recognize the Key Identity — Difference of Squares", "The core trick lies in recognizing that\n$$ \sqrt{n+1} - \sqrt{n} $$\nappears both as a numerator and a denominator after rationalization. This suggests a direct simplification, but first, focus on the first factor:\n$$ \frac{1}{\sqrt{n} + \sqrt{n+1}}. $$\nThe presence of the sum of square roots in the denominator calls for rationalization.", "---", "## Step 2: Rationalize the Denominator", "To simplify $ \displaystyle \frac{1}{\sqrt{n} + \sqrt{n+1}} $, multiply numerator and denominator by the conjugate of the denominator:\n$$ \sqrt{n+1} - \sqrt{n}. $$", "So,\n$$\n\frac{1}{\sqrt{n} + \sqrt{n+1}} \cdot \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} \n= \frac{\sqrt{n+1} - \sqrt{n}}{(\sqrt{n} + \sqrt{n+1})(\sqrt{n+1} - \sqrt{n})}\n$$", "---", "## Step 3: Apply the Difference of Squares Formula", "The denominator is now of the form $ (a + b)(a - b) = a^2 - b^2 $, with $ a = \sqrt{n+1} $, $ b = \sqrt{n} $:", "$$\n(\sqrt{n} + \sqrt{n+1})(\sqrt{n+1} - \sqrt{n}) = (\sqrt{n+1})^2 - (\sqrt{n})^2 = (n+1) - n = 1\n$$", "Thus, the entire expression simplifies to:\n$$\n\frac{\sqrt{n+1} - \sqrt{n}}{1} = \sqrt{n+1} - \sqrt{n}\n$$", "---", "## Why This Simplification Matters", "This identity is more than just an algebraic exercise:", "- Simplifies radicals in advanced mathematics and physics problems involving approximations or integrals.\n- Cleans computations in series evaluations (e.g., telescoping series like $ \sum(\sqrt{n+1} - \sqrt{n}) $).\n- Reinforces conjugate techniques, a fundamental skill in rationalizing denominators and solving fractional expressions.", "---", "## Final Result", "$$\n\boxed{ \frac{1}{\sqrt{n} + \sqrt{n+1}} \cdot \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} = \sqrt{n+1} - \sqrt{n} }\n$$", "By identifying and applying the conjugate rationalization and simplifying via the difference of squares, we effortlessly reduce a complex expression to its elegant simplest form.", "---", "### Takeaway\nWhen encountering expressions with radical denominators involving $ \sqrt{n} + \sqrt{n+1} $, remember: rationalize with the conjugate to unlock a clean result. This technique strengthens your algebra toolkit and supports deeper mathematical reasoning across many fields.", "---", "Keywords: Rationalizing denominators, difference of squares, simplify radicals, manipulation of square roots, algebraic identities, $ \sqrt{n+1} - \sqrt{n} $ simplification, fraction simplification.", "---", "Use this insight to improve your problem-solving and understand why this step-by-step approach works in algebra!"]