The value of the expression $\sqrt{4 + \sqrt{15}} + \sqrt{4 - \sqrt{15}} - \sqrt{12 - 4\sqrt{5}}$ is

Concept & Formula When dealing with a sum of square roots of conjugate binomial surds, setting the expression equal to $x$ and squaring both sides is highly efficient. For individual complex surds, convert them to perfect squares using $(a-b)^2 = a^2 + b^2 - 2ab$. $$ (a+b)^2 = a^2 + b^2 + 2ab $$ Step-by-Step Solution * Evaluate the first part of the expression: Let $x = \sqrt{4 + \sqrt{15}} + \sqrt{4 - \sqrt{15}}$ * Square both sides: $$ x^2 = (4 + \sqrt{15}) + (4 - \sqrt{15}) + 2\sqrt{(4 + \sqrt{15})(4 - \sqrt{15})} $$ $$ x^2 = 8 + 2\sqrt{16 - 15} $$ $$ x^2 = 8 + 2(1) = 10 $$ * Since $x$ must be positive, $x = \sqrt{10}$. * Evaluate the third term separately: $\sqrt{12 - 4\sqrt{5}}$ Rewrite $-4\sqrt{5}$ as $-2(2\sqrt{5}) = -2\sqrt{20}$. We need numbers adding to $12$ and multiplying to $20$. They are $10$ and $2$. So, $\sqrt{12 - 2\sqrt{20}} = \sqrt{10} - \sqrt{2}$ * Combine the components back into the original expression: $$ (\sqrt{10}) - (\sqrt{10} - \sqrt{2}) $$ $$ = \sqrt{10} - \sqrt{10} + \sqrt{2} = \sqrt{2} $$ * Analyze the result: $\sqrt{2}$ cannot be expressed as a simple fraction, meaning it is an irrational number. Exam Strategy & Shortcut Squaring a sum like $\sqrt{a+\sqrt{b}} + \sqrt{a-\sqrt{b}}$ will always yield $2a + 2\sqrt{a^2-b}$. Memorizing this shortcut lets you instantly compute $2(4) + 2\sqrt{16-15} = 8 + 2 = 10$, giving $\sqrt{10}$ for the first half in seconds, bypassing the manual algebraic expansion completely. Common Pitfall Attempting to rationalize or turn $4 + \sqrt{15}$ directly into a perfect square by multiplying and dividing by $2$. While mathematically sound, it leads to messy fractions (e.g., $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{2}}$) which increases the likelihood of arithmetic errors compared to the squaring method. Final Answer Therefore, the correct answer is an irrational number.