The square root of $$(7 + 3\sqrt{5})(7 - 3\sqrt{5})$$ is

### Concept & Formula Use the algebraic identity for the difference of squares: $$(a + b)(a - b) = a^2 - b^2$$ ### Step-by-Step Solution Let $a = 7$ and $b = 3\sqrt{5}$. Apply the identity to simplify the product inside the square root: $$(7 + 3\sqrt{5})(7 - 3\sqrt{5}) = 7^2 - (3\sqrt{5})^2$$ Calculate the squared values: * $7^2 = 49$ * $(3\sqrt{5})^2 = 3^2 \times 5 = 9 \times 5 = 45$ Subtract the values: $$49 - 45 = 4$$ The question asks for the square root of this product: $$\sqrt{4} = 2$$ ### Exam Strategy & Shortcut Recognize the difference of squares instantly: $49 - 9 \times 5 = 49 - 45 = 4$. The square root of $4$ is $2$. This entire computation can be done mentally within 5 seconds. ### Common Pitfall A common mistake is forgetting to take the square root at the end, leading to selecting option (c) $4$ instead of the actual square root, which is $2$. ### Final Answer **Therefore, the correct answer is 2.**