$$\sqrt{0.00004761}$$ equals

### Concept & Logic To evaluate a large decimal square root, split the task: find the square root of the significant digits, then adjust the decimal point based on the "half-rule" (the root has half the decimal places of the square). ### Step-by-Step Solution * **Given:** The expression $$\sqrt{0.00004761}$$ * **Calculation:** Step 1: Isolate the core integer, $4761$. Step 2: Estimate its square root. We know $70^2 = 4900$. So the root is slightly less than $70$. Since $4761$ ends in $1$, the root must end in $1$ or $9$. $69^2$ is the logical candidate. Let's verify: $69^2 = (70 - 1)^2 = 4900 - 140 + 1 = 4761$. So, $\sqrt{4761} = 69$. Step 3: Count the decimal places in the original number: $0.00004761$ has **$8$** decimal places. Step 4: The root will have $8 \div 2 = 4$ decimal places. Step 5: Apply $4$ decimal places to $69$. You need two leading zeros. $$0.0069$$ ### Exam Strategy & Shortcut Look at the options! All options use the digits $69$ or $609$. Since $609$ squared would be massive, the significant digits must be $69$. You just need to count the decimals. $8$ decimals in the question $\rightarrow$ $4$ decimals in the answer. Option (b) $0.0069$ is the only one with $4$ decimal places. You can solve this in $5$ seconds without actually calculating $69^2$. ### Common Pitfall * **Mistake:** Getting intimidated by large decimal numbers and trying to do long division method for square roots, which eats up minutes. * **Correction:** Leverage the multiple-choice format to bypass integer calculations and rely solely on decimal counting rules. ### Final Answer **Therefore, the correct answer is 0.0069.**