If $\sqrt{5} = 2.236$, then the value of $\frac{\sqrt{5}}{2} - \frac{10}{\sqrt{5}} + \sqrt{125}$ is equal to

### Concept & Strategy When confronted with an expression containing multiple surds of the same base, simplify and rationalize every term so they all contain a common radical. This allows for straightforward addition and subtraction. ### Step-by-Step Solution Given: $\sqrt{5} = 2.236$ We need to evaluate the expression: $\frac{\sqrt{5}}{2} - \frac{10}{\sqrt{5}} + \sqrt{125}$ Simplify each term individually to have $\sqrt{5}$ as a common factor. **Term 1:** $\frac{\sqrt{5}}{2}$ remains as is (or can be written as $0.5\sqrt{5}$). **Term 2:** Rationalize $\frac{10}{\sqrt{5}}$ by multiplying top and bottom by $\sqrt{5}$: $\frac{10\sqrt{5}}{5} = 2\sqrt{5}$ **Term 3:** Simplify $\sqrt{125}$ by extracting the largest perfect square: $\sqrt{25 \times 5} = 5\sqrt{5}$ Substitute these simplified terms back into the original expression: $0.5\sqrt{5} - 2\sqrt{5} + 5\sqrt{5}$ Combine the coefficients: $(0.5 - 2 + 5)\sqrt{5} = 3.5\sqrt{5}$ Now, substitute the given value for $\sqrt{5}$: $3.5 \times 2.236$ Calculate the final product: $3.5 \times 2.236 = 7.826$ ### Exam Strategy & Shortcut Factor out the $\sqrt{5}$ early on. The expression is $\sqrt{5} \times (\frac{1}{2} - \frac{10}{5} + \sqrt{25})$. This simplifies instantly to $\sqrt{5} \times (0.5 - 2 + 5) = 3.5 \times \sqrt{5}$. To multiply $3.5$ quickly, multiply $2.236$ by $3$ ($6.708$) and add half of $2.236$ ($1.118$). $6.708 + 1.118 = 7.826$. ### Common Pitfall Failing to simplify $\sqrt{125}$ properly or trying to evaluate $10 \div 2.236$ manually. Always clean up the algebraic roots before plugging in the decimal values. ### Final Answer Therefore, the correct answer is **7.826**.