$$\sqrt{\frac{25}{81} - \frac{1}{9}} = x$$

### Concept & Formula To simplify an expression under a square root involving fractions, find a common denominator to subtract the fractions first, then take the square root of the resulting fraction. $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ ### Step-by-Step Solution * **Given:** The expression $$\sqrt{\frac{25}{81} - \frac{1}{9}}$$ * **Calculation:** Find the common denominator for $81$ and $9$, which is $81$. Multiply the numerator and denominator of $\frac{1}{9}$ by $9$: $$\frac{1}{9} = \frac{9}{81}$$ Subtract the fractions inside the radical: $$\frac{25}{81} - \frac{9}{81} = \frac{16}{81}$$ Take the square root: $$\sqrt{\frac{16}{81}} = \frac{\sqrt{16}}{\sqrt{81}} = \frac{4}{9}$$ ### Exam Strategy & Shortcut Recognize perfect squares immediately. The numbers $25, 81, 9, 16$ are all perfect squares. As soon as you compute the numerator subtraction $25 - 9 = 16$, you know the answer must be $\sqrt{\frac{16}{81}} = \frac{4}{9}$ without writing down intermediate steps. ### Common Pitfall * **Mistake:** Distributing the square root over subtraction, i.e., mistakenly computing $\sqrt{\frac{25}{81}} - \sqrt{\frac{1}{9}} = \frac{5}{9} - \frac{1}{3} = \frac{2}{9}$. * **Correction:** Remember that $\sqrt{A - B} \neq \sqrt{A} - \sqrt{B}$. Always perform the subtraction inside the radical first. ### Final Answer **Therefore, the correct answer is 4/9.**