$\sqrt{6 - 4\sqrt{3} + \sqrt{16 - 8\sqrt{3}}}$ is equal to

Concept & Formula This problem involves nested square roots. The strategy is to simplify from the innermost square root outwards by turning binomial surds into perfect squares using the identity: $$ (a - b)^2 = a^2 + b^2 - 2ab $$ Step-by-Step Solution * Start with the innermost root: $\sqrt{16 - 8\sqrt{3}}$ * Convert the term $-8\sqrt{3}$ to the form $-2ab$. $$ 16 - 8\sqrt{3} = 16 - 2(4\sqrt{3}) = 16 - 2\sqrt{16 \times 3} = 16 - 2\sqrt{48} $$ * We need two numbers that add to $16$ and multiply to $48$. These are $12$ and $4$. * Rewrite: $$ (\sqrt{12})^2 + (\sqrt{4})^2 - 2\sqrt{12}\sqrt{4} = (\sqrt{12} - \sqrt{4})^2 $$ * Simplify $\sqrt{12} - \sqrt{4}$ to $2\sqrt{3} - 2$. * So, the innermost root evaluates to $2\sqrt{3} - 2$. (Note: $2\sqrt{3} > 2$, so it is positive). * Substitute this back into the main expression: $$ \sqrt{6 - 4\sqrt{3} + (2\sqrt{3} - 2)} $$ * Combine like terms: $$ \sqrt{4 - 2\sqrt{3}} $$ * Now, simplify this new root. We need two numbers adding to $4$ and multiplying to $3$. They are $3$ and $1$. $$ 4 - 2\sqrt{3} = 3 + 1 - 2\sqrt{3} = (\sqrt{3} - 1)^2 $$ * Taking the square root gives $\sqrt{3} - 1$. Exam Strategy & Shortcut For nested surds, always work from the inside out. When simplifying $16 - 8\sqrt{3}$, instead of pushing the whole $4$ into the root to make $\sqrt{48}$, you can mentally break down $8\sqrt{3}$ as $2 \times 4 \times \sqrt{3}$. Ask yourself: Does $4^2 + (\sqrt{3})^2 = 16$? Yes! So it's immediately $(4 - \sqrt{3})^2$. No, wait. $16 + 3 = 19$, not $16$. Let's break it differently: $2 \times (2\sqrt{3}) \times 2$. Does $(2\sqrt{3})^2 + 2^2 = 16$? $12 + 4 = 16$. Yes! So the root is $2\sqrt{3} - 2$. This mental check is much faster than full expansion. Common Pitfall When taking the square root of $(x - y)^2$, always ensure $x > y$ so the result is positive. For instance, if you incorrectly write $\sqrt{4 - 2\sqrt{3}}$ as $1 - \sqrt{3}$, you will get a negative value, which violates the principle of real square roots. Final Answer Therefore, the correct answer is $\sqrt{3} - 1$.