If $\sqrt{33} = 5.745$, then the value of the following is approximately $\sqrt{\frac{3}{11}}$

Concept & Formula To find the value of a fraction under a square root when given the root of a related product, rationalize the denominator of the fraction inside the square root to match the given information. $$ \sqrt{\frac{a}{b}} = \frac{\sqrt{a \times b}}{b} $$ Step-by-Step Solution * **Given:** $\sqrt{33} = 5.745$ and we need to evaluate $\sqrt{\frac{3}{11}}$ * **Calculation:** To utilize the given value of $\sqrt{33}$, multiply the numerator and the denominator of the fraction inside the root by 11. $\sqrt{\frac{3}{11}} = \sqrt{\frac{3 \times 11}{11 \times 11}}$ * Simplify the expression: $= \sqrt{\frac{33}{121}}$ * Apply the square root to the numerator and denominator separately: $= \frac{\sqrt{33}}{\sqrt{121}}$ $= \frac{\sqrt{33}}{11}$ * Substitute the given value: $= \frac{5.745}{11}$ * Perform the division: $5.745 \div 11 = 0.52227...$ * Round to four decimal places as seen in the options: $\approx 0.5223$ Exam Strategy & Shortcut Whenever you are given the value of $\sqrt{a \times b}$ and asked for $\sqrt{a/b}$, the fastest shortcut is to multiply the top and bottom by $b$. This immediately gives you $\frac{\sqrt{ab}}{b}$. Here, it turns $\sqrt{3/11}$ into $\frac{\sqrt{33}}{11}$, turning a complex square root problem into a simple division problem. Common Pitfall A common mistake is trying to independently estimate the values of $\sqrt{3}$ and $\sqrt{11}$ and divide them. While this works mathematically, it is highly prone to rounding errors and takes significantly more time than using the provided value of $\sqrt{33}$. Final Answer **Therefore, the correct answer is 0.5223.**