If $3\sqrt{5} + \sqrt{125} = 17.88$, then what will be the value of $\sqrt{80} + 6\sqrt{5}$ ?

### Concept & Logic When given the decimal value of an initial surd equation, simplify both the given equation and the target expression down to the same base surd. This allows you to find the decimal equivalent of the base surd and substitute it into the final expression. ### Step-by-Step Solution * Given Equation: $3\sqrt{5} + \sqrt{125} = 17.88$ * Simplify the given equation to isolate the base surd ($\sqrt{5}$): * $\sqrt{125} = \sqrt{25 \times 5} = 5\sqrt{5}$ * $3\sqrt{5} + 5\sqrt{5} = 17.88$ * $8\sqrt{5} = 17.88$ * Solve for $\sqrt{5}$: * $\sqrt{5} = \frac{17.88}{8}$ * $\sqrt{5} = 2.235$ * Now, simplify the target expression: $\sqrt{80} + 6\sqrt{5}$ * $\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$ * $(4\sqrt{5}) + 6\sqrt{5} = 10\sqrt{5}$ * Substitute the decimal value of $\sqrt{5}$ into the simplified target expression: * $10\sqrt{5} = 10 \times 2.235 = 22.35$ ### Exam Strategy & Shortcut Instead of calculating the exact value of $\sqrt{5}$, you can use ratios. The given expression equals $8\sqrt{5}$ and the target expression equals $10\sqrt{5}$. This means the target is $\frac{10}{8}$ (or 1.25 times) the original amount. Multiply the given value by 1.25 directly: $17.88 \times 1.25 = 17.88 + 4.47 = 22.35$. ### Common Pitfall Avoid calculating the square root of 5 manually or rounding the intermediate step (like rounding 2.235 to 2.2). Rounding too early will cascade into a larger error in the final multiplication, causing you to pick an incorrect close option. ### Final Answer **Therefore, the correct answer is 22.35.**