$\sqrt{\frac{16}{25}} \times \sqrt{\frac{x}{25}} \times \frac{16}{25} = \frac{256}{625}$

### Concept & Strategy Simplify algebraic fraction chains by isolating the radical element containing the variable and cancelling matching factors from both sides of the equation. Simplification rule: $$ \text{If } a \times \sqrt{x} = b, \text{ then } \sqrt{x} = \frac{b}{a} $$ ### Step-by-Step Solution - **Step 1: Simplify known square roots and separate variables.** - $\sqrt{\frac{16}{25}} = \frac{4}{5}$ - $\sqrt{\frac{x}{25}} = \frac{\sqrt{x}}{5}$ - **Step 2: Substitute expressions back into the primary equation.** - $\frac{4}{5} \times \frac{\sqrt{x}}{5} \times \frac{16}{25} = \frac{256}{625}$ - Combine constant multipliers on the left: $\frac{4 \times 16}{5 \times 5 \times 25} \times \sqrt{x} = \frac{64}{625} \times \sqrt{x}$ - Equation is now: $\frac{64}{625} \times \sqrt{x} = \frac{256}{625}$ - **Step 3: Cancel denominators and solve for $\sqrt{x}$.** - Multiply both sides by $625$: $64 \times \sqrt{x} = 256$ - $\sqrt{x} = \frac{256}{64} = 4$ - **Step 4: Square both sides to find $x$.** - $x = 4^2 = 16$ ### Exam Strategy & Shortcut Notice that the right hand side is $\frac{256}{625} = \left(\frac{16}{25}\right)^2$. The expression on the left is $\frac{4}{5} \times \sqrt{\frac{x}{25}} \times \frac{16}{25}$. Divide out $\frac{16}{25}$ from both sides to get: $\frac{4}{5} \times \sqrt{\frac{x}{25}} = \frac{16}{25}$. We know $\frac{4}{5} \times \frac{4}{5} = \frac{16}{25}$, so $\sqrt{\frac{x}{25}} = \frac{4}{5}$. This means $\frac{x}{25} = \frac{16}{25}$, revealing directly that $x = 16$. ### Common Pitfall Getting confused by the multiple repetitions of numbers like $16$ and $25$ throughout the problem structure. Work linearly and keep track of which components reside inside a radical and which do not. ### Final Answer Therefore, the correct answer is 16.