What is $\frac{5+\sqrt{10}}{5\sqrt{5}-2\sqrt{20}-\sqrt{32}+\sqrt{50}}$ equal to

Concept & Formula To simplify complex radical fractions, you must break down every surd into its simplest form by extracting perfect squares. Once the denominator is simplified, look for common factors to factor out in both the numerator and denominator to cancel terms. $$ \sqrt{a^2b} = a\sqrt{b} $$ Step-by-Step Solution * **Given:** $\frac{5+\sqrt{10}}{5\sqrt{5}-2\sqrt{20}-\sqrt{32}+\sqrt{50}}$ * **Calculation:** First, simplify each term in the denominator independently: $5\sqrt{5}$ (Already simplified) $2\sqrt{20} = 2\sqrt{4 \times 5} = 2(2\sqrt{5}) = 4\sqrt{5}$ $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$ $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ * Reconstruct and simplify the denominator: $= 5\sqrt{5} - 4\sqrt{5} - 4\sqrt{2} + 5\sqrt{2}$ Group like terms: $= (5 - 4)\sqrt{5} + (-4 + 5)\sqrt{2}$ $= \sqrt{5} + \sqrt{2}$ * The full expression is now: $\frac{5+\sqrt{10}}{\sqrt{5}+\sqrt{2}}$ * Now, look at the numerator. Factor out $\sqrt{5}$: Notice that $5 = \sqrt{25} = \sqrt{5} \times \sqrt{5}$ and $\sqrt{10} = \sqrt{5} \times \sqrt{2}$. $5 + \sqrt{10} = \sqrt{5}(\sqrt{5} + \sqrt{2})$ * Substitute this back into the fraction: $= \frac{\sqrt{5}(\sqrt{5} + \sqrt{2})}{\sqrt{5}+\sqrt{2}}$ * Cancel the common binomial term $(\sqrt{5} + \sqrt{2})$ from the top and bottom: $= \sqrt{5}$ Exam Strategy & Shortcut Don't immediately jump to rationalizing the denominator. Rationalizing $\sqrt{5} + \sqrt{2}$ by multiplying by its conjugate $\sqrt{5} - \sqrt{2}$ works, but it takes an extra step. In exam settings, numerators are almost always designed to factor cleanly into a multiple of the simplified denominator. Always check for a common factor first! Common Pitfall A common mistake is incorrectly simplifying the surds (e.g., extracting the wrong perfect square) which prevents the denominator from collapsing into a simple binomial. Always double-check your prime factorizations for numbers like 20, 32, and 50. Final Answer **Therefore, the correct answer is \sqrt{5}.**