$\left(\sqrt{\frac{225}{729}} - \sqrt{\frac{25}{144}}\right) \div \sqrt{\frac{16}{81}} = x$

### Concept & Formula Simplify fractions containing perfect squares by evaluating the square root of the numerator and denominator separately before executing the arithmetic operations. Radical rule for division: $$ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $$ ### Step-by-Step Solution - **Step 1: Evaluate each individual square root.** - $\sqrt{\frac{225}{729}} = \frac{15}{27} = \frac{5}{9}$ - $\sqrt{\frac{25}{144}} = \frac{5}{12}$ - $\sqrt{\frac{16}{81}} = \frac{4}{9}$ - **Step 2: Simplify the expression inside the parentheses.** - We have $\left(\frac{5}{9} - \frac{5}{12}\right)$. - Find the LCM of $9$ and $12$, which is $36$. - $\frac{5 \times 4}{36} - \frac{5 \times 3}{36} = \frac{20 - 15}{36} = \frac{5}{36}$ - **Step 3: Perform the division.** - $\frac{5}{36} \div \frac{4}{9} = \frac{5}{36} \times \frac{9}{4}$ - $\frac{5}{4 \times 4} = \frac{5}{16}$ ### Exam Strategy & Shortcut Simplify fractions at the very first step. Reducing $\frac{15}{27}$ directly to $\frac{5}{9}$ minimizes the LCM computation from $108$ down to $36$. When dividing fractions, instantly multiply by the reciprocal to keep your scratch work clean and swift. ### Common Pitfall Forgetting to invert the divisor when changing from a division operator to a multiplication operator, which leads to computing $\frac{5}{36} \times \frac{4}{9} = \frac{5}{81}$ instead of the correct answer. ### Final Answer Therefore, the correct answer is 5/16.