What should come in place of both the question marks in the equation $\frac{x}{\sqrt{128}} = \frac{\sqrt{162}}{x}$.

Concept & Formula When an unknown variable appears diagonally across an equals sign in a proportion, cross-multiplying allows you to isolate the variable squared. $$a \times a = b \times c \implies a^2 = bc$$ Step-by-Step Solution * **Given:** $\frac{x}{\sqrt{128}} = \frac{\sqrt{162}}{x}$ * **Calculation:** Cross-multiply to eliminate the fractions: $x \cdot x = \sqrt{128} \cdot \sqrt{162}$ $x^2 = \sqrt{128 \times 162}$ * To avoid massive multiplication, break the numbers down into prime factors or recognizable perfect squares: $128 = 64 \times 2$ $162 = 81 \times 2$ * Substitute these back into the root: $x^2 = \sqrt{(64 \times 2) \times (81 \times 2)}$ $x^2 = \sqrt{64 \times 81 \times 4}$ * Extract the square root of each perfect square: $x^2 = 8 \times 9 \times 2$ $x^2 = 144$ * Take the square root of both sides to find $x$: $x = \sqrt{144} = 12$ Exam Strategy & Shortcut Instead of fully calculating $\sqrt{128 \times 162}$, look at the units digit. The last digit of $128 \times 162$ will end in $6$ ($8 \times 2$). The square root of a number ending in $6$ must end in $4$ or $6$. The options don't directly help here since it is inside a secondary root, but breaking it into $64$ and $81$ (which are highly recognizable perfect squares) is the fastest mental math path. Common Pitfall Multiplying $128 \times 162$ to get $20736$ and then wasting immense time trying to calculate the square root of a 5-digit number without a calculator. Always factorize first. Final Answer **Therefore, the correct answer is 12.**