If $\sqrt{5} = 2.236$, then the value of $\frac{1}{\sqrt{5}}$ is

### Concept & Strategy When dividing by an irrational number like a square root, always rationalize the denominator first. This transforms complex division into simple multiplication or division by an integer. $$ \frac{a}{\sqrt{b}} = \frac{a \times \sqrt{b}}{b} $$ ### Step-by-Step Solution Given: $\sqrt{5} = 2.236$ We need to evaluate the expression: $\frac{1}{\sqrt{5}}$ Rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{5}$: $\frac{1}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5}$ Substitute the given value for $\sqrt{5}$: $\frac{2.236}{5}$ Perform the simple division: $2.236 \div 5 = 0.4472$ Rounding to three decimal places to match the options, we get $0.447$. ### Exam Strategy & Shortcut Instead of dividing by $5$, you can multiply the numerator by $2$ and divide by $10$. $2.236 \times 2 = 4.472$. Dividing by $10$ moves the decimal point one place to the left, yielding $0.4472$. This is much faster mental math. ### Common Pitfall The most common trap is attempting to divide $1$ directly by the decimal $2.236$ (i.e., $1 / 2.236$). This leads to a messy, time-consuming calculation that easily results in errors. ### Final Answer Therefore, the correct answer is **.447**.