s\sqrt{2} = 10 \Rightarrow s = \frac{10}{\sqrt{2}} = 5\sqrt{2}

["# Solving the Equation: s√2 = 10 and Simplifying to s = 5√2", "Mathematics often presents elegant solutions rooted in square roots and rational numbers. One classic example is the equation:", "[\ns\sqrt{2} = 10\n]", "This equation may seem simple, but it illustrates a powerful technique for isolating variables involving irrational numbers — specifically, square roots. Let’s explore how this equation is solved step-by-step and why the final simplified form is:", "[\ns = \frac{10}{\sqrt{2}} = 5\sqrt{2}\n]", "---", "## Understanding the Equation", "We start with the equation:", "[\ns\sqrt{2} = 10\n]", "Here, ( s ) is multiplied by ( \sqrt{2} ), an irrational number approximately equal to 1.414. To solve for ( s ), we divide both sides by ( \sqrt{2} ) to isolate ( s ):", "[\ns = \frac{10}{\sqrt{2}}\n]", "However, it’s conventional to rationalize the denominator to express the result in its simplest, most elegant form.", "---", "## Rationalizing the Denominator", "Dividing by ( \sqrt{2} ) gives a fractional expression involving an irrational denominator:", "[\n\frac{10}{\sqrt{2}}\n]", "To rationalize this, multiply both the numerator and denominator by ( \sqrt{2} ):", "[\ns = \frac{10 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}\n]", "This step not only simplifies the expression but also reveals a deeper mathematical truth — expressions involving square roots can be simplified into rationalized forms involving clean radicals, improving clarity and usability in further calculations.", "---", "## Final Simplified Value", "Thus, the exact value of ( s ) satisfying ( s\sqrt{2} = 10 ) is:", "[\ns = 5\sqrt{2}\n]", "This form is preferred in algebra because:\n- It eliminates irrational denominators.\n- It highlights the irrational component ( \sqrt{2} ) in its simplest product form.\n- It facilitates easier computation, especially in trigonometry, geometry, and complex equation solving.", "---", "## Applications and Why It Matters", "Equations like ( s\sqrt{2} = 10 ) often appear in trigonometric identities, geometric constructions (e.g., diagonal lengths in squares), and physics problems involving vectors or periodic phenomena.", "Understanding how to manipulate and simplify such equations ensures accuracy and efficiency in mathematical modeling and problem-solving. The rationalized form ( s = 5\sqrt{2} ) is not just an answer — it’s a precise, elegant representation that honors the nature of irrational numbers.", "---", "## Conclusion", "From ( s\sqrt{2} = 10 ) to its simplified radical form ( s = 5\sqrt{2} ), this process exemplifies the beauty of algebra: transforming complexity into clarity through rationalization. Whether for homework, exams, or real-world applications, mastering these steps strengthens mathematical fluency and appreciation for the quiet power of radicals.", "Key takeaway: Always simplify radicals by rationalizing the denominator for a cleaner, more useful result — and remember:", "[\ns = \frac{10}{\sqrt{2}} = 5\sqrt{2}\n]", "Happy calculating!"]