The square root of `(0.342 xx 0.684)/(0.000342 xx 0.000171)` is :

Options

A. 250
B. 2500
C. 2000
D. 4000

Answer

To solve the expression \(\sqrt{\frac{0.342 \times 0.684}{0.000342 \times 0.000171}}\), we will simplify the terms step by step.

### Step 1: Rewrite the numbers in scientific notation
First, we convert the decimal numbers into scientific notation for easier manipulation.

- \(0.342 = 3.42 \times 10^{-1}\)
- \(0.684 = 6.84 \times 10^{-1}\)
- \(0.000342 = 3.42 \times 10^{-4}\)
- \(0.000171 = 1.71 \times 10^{-4}\)

### Step 2: Substitute the scientific notation into the expression
Now, we substitute these values into the expression:

\[ \sqrt{\frac{(3.42 \times 10^{-1}) \times (6.84 \times 10^{-1})}{(3.42 \times 10^{-4}) \times (1.71 \times 10^{-4})}} \]

### Step 3: Simplify the numerator and the denominator
Now, we simplify the numerator and the denominator separately.

**Numerator:**
\[ 3.42 \times 6.84 = 23.328 \]
\[ 10^{-1} \times 10^{-1} = 10^{-2} \]
So, the numerator becomes:
\[ 23.328 \times 10^{-2} \]

**Denominator:**
\[ 3.42 \times 1.71 = 5.8472 \]
\[ 10^{-4} \times 10^{-4} = 10^{-8} \]
So, the denominator becomes:
\[ 5.8472 \times 10^{-8} \]

### Step 4: Combine the numerator and denominator
Now, we can write the entire expression as:
\[ \sqrt{\frac{23.328 \times 10^{-2}}{5.8472 \times 10^{-8}}} \]

### Step 5: Simplify the fraction
This can be simplified to:
\[ \frac{23.328}{5.8472} \times 10^{(-2) - (-8)} = \frac{23.328}{5.8472} \times 10^{6} \]

### Step 6: Calculate the fraction
Calculating \( \frac{23.328}{5.8472} \):
\[ \frac{23.328}{5.8472} \approx 3.99 \]

So, we have:
\[ 3.99 \times 10^{6} \]

### Step 7: Take the square root
Now, we take the square root of the entire expression:
\[ \sqrt{3.99 \times 10^{6}} = \sqrt{3.99} \times \sqrt{10^{6}} = \sqrt{3.99} \times 10^{3} \]

Calculating \( \sqrt{3.99} \approx 1.9975 \), we have:
\[ 1.9975 \times 10^{3} \approx 1997.5 \]

### Final Answer
Thus, the final answer is approximately \(2000\) (rounded).

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