The approximate value of $\frac{3\sqrt{12}}{2\sqrt{28}} \div \frac{2\sqrt{21}}{\sqrt{98}}$ is

### Concept & Formula When dividing fractions containing surds, flip the second fraction (multiply by its reciprocal) and break down all composite numbers under the roots into prime factors to allow for massive cross-cancellation. $$ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} $$ ### Step-by-Step Solution * Rewrite the division as multiplication by inverting the second fraction: * $\frac{3\sqrt{12}}{2\sqrt{28}} \times \frac{\sqrt{98}}{2\sqrt{21}}$ * Simplify all surds by pulling out perfect squares: * $\sqrt{12} = 2\sqrt{3}$ * $\sqrt{28} = 2\sqrt{7}$ * $\sqrt{98} = 7\sqrt{2}$ * $\sqrt{21} = \sqrt{3}\sqrt{7}$ (leave as prime factors for cancellation) * Substitute simplified surds into the expression: * $\frac{3(2\sqrt{3})}{2(2\sqrt{7})} \times \frac{7\sqrt{2}}{2(\sqrt{3}\sqrt{7})}$ * $= \frac{6\sqrt{3}}{4\sqrt{7}} \times \frac{7\sqrt{2}}{2\sqrt{3}\sqrt{7}}$ * Cancel common terms from numerators and denominators: * The $\sqrt{3}$ in the top left cancels with the $\sqrt{3}$ in the bottom right. * Combine the remaining terms: $\frac{6 \times 7\sqrt{2}}{4 \times 2 \times (\sqrt{7} \times \sqrt{7})}$ * Simplify the resulting fraction: * $\sqrt{7} \times \sqrt{7} = 7$. * $= \frac{42\sqrt{2}}{8 \times 7} = \frac{42\sqrt{2}}{56}$ * Divide top and bottom by 14: $= \frac{3\sqrt{2}}{4}$ * Calculate the final decimal value using $\sqrt{2} \approx 1.414$: * $\frac{3 \times 1.414}{4} = \frac{4.242}{4} = 1.0605$ ### Exam Strategy & Shortcut Write everything as single large roots to cancel internal factors before pulling out perfect squares. $\frac{3}{2} \times \frac{1}{2} \times \sqrt{\frac{12 \times 98}{28 \times 21}}$ Cancel 7 from 98 and 28: $\sqrt{\frac{12 \times 14}{4 \times 21}}$ Cancel 4 from 12 and 4: $\sqrt{\frac{3 \times 14}{21}}$ Cancel 3 from 3 and 21: $\sqrt{\frac{14}{7}} = \sqrt{2}$. The whole expression elegantly collapses to $\frac{3}{4} \times \sqrt{2}$. Much faster than extracting individual perfect squares! ### Common Pitfall Forgetting to flip the second fraction when converting division to multiplication is a classic error. It leads you to multiply numerators with numerators, leaving an impossible tangle of roots that won't cancel out properly. ### Final Answer **Therefore, the correct answer is 1.0605.**