$$\sqrt{\frac{0.081 \times 0.324 \times 4.624}{1.5625 \times 0.0289 \times 72.9 \times 64}}$$ is equal to

### Concept & Formula Eliminate the decimal points by counting the total number of decimal places in the numerator and the denominator. Once converted to integers, simplify the perfect squares. ### Step-by-Step Solution Count the total decimal places: * Numerator: $3 \text{ (from 0.081)} + 3 \text{ (from 0.324)} + 3 \text{ (from 4.624)} = 9$ decimal places. * Denominator: $4 \text{ (from 1.5625)} + 4 \text{ (from 0.0289)} + 1 \text{ (from 72.9)} = 9$ decimal places. Since the decimal places balance out, remove the decimals entirely: $$\sqrt{\frac{81 \times 324 \times 4624}{15625 \times 289 \times 729 \times 64}}$$ Find the square roots of the individual perfect squares: * $\sqrt{81} = 9$ * $\sqrt{324} = 18$ * $\sqrt{4624} = 68$ * $\sqrt{15625} = 125$ * $\sqrt{289} = 17$ * $\sqrt{729} = 27$ * $\sqrt{64} = 8$ Substitute the square roots back into the expression: $$\frac{9 \times 18 \times 68}{125 \times 17 \times 27 \times 8}$$ Simplify the expression by cancellation: * $\frac{9}{27} = \frac{1}{3}$ * $\frac{18}{3} = 6$ * $\frac{68}{17} = 4$ * $\frac{4}{8} = \frac{1}{2}$ * $\frac{6}{2} = 3$ This leaves us with: $$\frac{3}{125} = \frac{3 \times 8}{125 \times 8} = \frac{24}{1000} = 0.024$$ ### Exam Strategy & Shortcut Instead of fully calculating, look at the denominator's dominant term $\sqrt{15625} = 125$. Any fraction with $125$ in the denominator converted to a decimal will be a multiple of $0.008$. Since $3/125 = 24/1000 = 0.024$, you can quickly match the option ending in $24$ with the correct decimal placement. ### Common Pitfall Misidentifying the square root of $4624$ or $729$ can lead to wrong values. Memorizing squares up to $30$ and practicing base-method estimation for larger squares helps prevent these calculation errors. ### Final Answer **Therefore, the correct answer is 0.024.**