$\sqrt{8 - 2\sqrt{15}}$ is equal to

Concept & Formula To find the square root of a surd in the form $a - 2\sqrt{b}$, we express the term inside the root as a perfect square $(x - y)^2$. $$ (x - y)^2 = x^2 + y^2 - 2xy $$ Step-by-Step Solution * Given: $\sqrt{8 - 2\sqrt{15}}$ * We need to express $8 - 2\sqrt{15}$ as $(\sqrt{m} - \sqrt{n})^2$. * We need two numbers $m$ and $n$ such that: $$ m + n = 8 $$ $$ m \times n = 15 $$ * The two numbers that satisfy both conditions are $5$ and $3$. * Rewrite the expression inside the square root: $$ 8 - 2\sqrt{15} = 5 + 3 - 2\sqrt{15} $$ $$ = (\sqrt{5})^2 + (\sqrt{3})^2 - 2(\sqrt{5})(\sqrt{3}) $$ * This is exactly the expansion of $(\sqrt{5} - \sqrt{3})^2$. * Taking the square root of the perfect square: $$ \sqrt{(\sqrt{5} - \sqrt{3})^2} = |\sqrt{5} - \sqrt{3}| $$ * Since $\sqrt{5} > \sqrt{3}$, the value is positive: $\sqrt{5} - \sqrt{3}$. Exam Strategy & Shortcut The fastest way to solve this is mental factoring. When you see $a - 2\sqrt{b}$, immediately think "factors of $b$ that add to $a$." Factors of $15$ adding to $8$ are $5$ and $3$. The sign in the middle is minus, so the answer is just $\sqrt{5} - \sqrt{3}$. Common Pitfall A common error is putting the smaller number first, resulting in $\sqrt{3} - \sqrt{5}$. A principal square root must be non-negative, so you must always subtract the smaller surd from the larger surd when extracting the root. Final Answer Therefore, the correct answer is $\sqrt{5} - \sqrt{3}$.