The value of `sqrt((47/5))` is -

Options

A. `cdot32`
B. `3cdot17`
C. `3cdot06`
D. None of these

Answer

To find the value of \( \sqrt{\frac{47}{5}} \), we will follow these steps:

### Step 1: Simplify the Fraction
First, we need to simplify the fraction inside the square root:
\[ \frac{47}{5} = 9.4 \]

### Step 2: Calculate the Square Root
Next, we will find the square root of \( 9.4 \):
\[ \sqrt{9.4} \]

### Step 3: Estimate the Square Root
To estimate \( \sqrt{9.4} \), we can find perfect squares around it. We know:
\[ 3^2 = 9 \quad \text{and} \quad 4^2 = 16 \]
Thus, \( \sqrt{9.4} \) is between 3 and 4.

### Step 4: Refine the Estimate
To refine our estimate, we can try \( 3.1 \) and \( 3.2 \):
\[ 3.1^2 = 9.61 \quad \text{(too high)} \]
\[ 3.2^2 = 10.24 \quad \text{(too high)} \]
So, we know \( \sqrt{9.4} \) is slightly less than \( 3.1 \).

### Step 5: Use Long Division Method for Square Roots
We can also use the long division method to find \( \sqrt{9.4} \) more accurately.

1. Pair the digits: \( 9.40 \) becomes \( 9 | 40 \).
2. Find the largest square less than or equal to \( 9 \), which is \( 3^2 = 9 \).
3. Subtract \( 9 \) from \( 9 \) to get \( 0 \).
4. Bring down \( 40 \) to get \( 040 \).
5. Double the current result (which is \( 3 \)) to get \( 6 \) and find a digit \( d \) such that \( 6d \times d \) is less than or equal to \( 40 \).
6. Trying \( d = 6 \): \( 66 \times 6 = 396 \) (too high).
7. Trying \( d = 5 \): \( 65 \times 5 = 325 \) (too high).
8. Trying \( d = 4 \): \( 64 \times 4 = 256 \) (too high).
9. Trying \( d = 3 \): \( 63 \times 3 = 189 \) (too high).
10. Trying \( d = 2 \): \( 62 \times 2 = 124 \) (too high).
11. Trying \( d = 1 \): \( 61 \times 1 = 61 \) (too high).
12. Trying \( d = 0 \): \( 60 \times 0 = 0 \) (valid).

### Step 6: Final Calculation
Continuing this process, we can find that \( \sqrt{9.4} \approx 3.07 \) after a few more iterations.

### Conclusion
Thus, the value of \( \sqrt{\frac{47}{5}} \) is approximately \( 3.07 \).

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