$\frac{\sqrt{24} + \sqrt{216}}{\sqrt{96}} = x$

### Concept & Formula When dealing with a fraction containing square roots in both the numerator and denominator, simplify all surds to their base forms first. This often reveals a common factor that can be canceled out. ### Step-by-Step Solution * Simplify each term in the fraction by finding the largest perfect square factors: * $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$ * $\sqrt{216} = \sqrt{36 \times 6} = 6\sqrt{6}$ * $\sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}$ * Substitute these simplified forms back into the original fraction: * $x = \frac{2\sqrt{6} + 6\sqrt{6}}{4\sqrt{6}}$ * Combine the like terms in the numerator: * $x = \frac{(2 + 6)\sqrt{6}}{4\sqrt{6}}$ * $x = \frac{8\sqrt{6}}{4\sqrt{6}}$ * Cancel out the common $\sqrt{6}$ term from the numerator and denominator, and divide the integers: * $x = \frac{8}{4} = 2$ ### Exam Strategy & Shortcut Notice that all the numbers inside the square roots (24, 216, 96) share a common factor of 24. $\sqrt{216} = \sqrt{9 \times 24} = 3\sqrt{24}$ $\sqrt{96} = \sqrt{4 \times 24} = 2\sqrt{24}$ Using this, the expression immediately becomes $\frac{\sqrt{24} + 3\sqrt{24}}{2\sqrt{24}} = \frac{4\sqrt{24}}{2\sqrt{24}} = 2$. This bypasses breaking them all the way down to prime factors. ### Common Pitfall Students sometimes try to cancel terms directly across addition without factoring first (e.g., trying to divide 24 by 96). You must always add or subtract the terms in the numerator to form a single product before canceling with the denominator. ### Final Answer **Therefore, the correct answer is 2.**