Find the square root of `105(4)/(64)`.

Options

A. `15(1)/(4)`
B. `15(12)/(4)`
C. `10(1)/(4)`
D. `6(2)/(4)`

Answer

To find the square root of \( \frac{1054}{64} \), we can follow these steps:

### Step 1: Simplify the fraction
We start with the expression:
\[ \frac{1054}{64} \]

### Step 2: Find the square root of the numerator and the denominator separately
We can separate the square root of the fraction:
\[ \sqrt{\frac{1054}{64}} = \frac{\sqrt{1054}}{\sqrt{64}} \]

### Step 3: Calculate the square root of the denominator
The square root of \( 64 \) is:
\[ \sqrt{64} = 8 \]

### Step 4: Calculate the square root of the numerator
Next, we need to find \( \sqrt{1054} \). We can factor \( 1054 \):
\[ 1054 = 2 \times 527 \]
Now, we can check if \( 527 \) can be factored further. It can be factored as:
\[ 527 = 17 \times 31 \]
Thus, \( 1054 = 2 \times 17 \times 31 \).

Since \( 1054 \) does not have any perfect square factors, we can approximate \( \sqrt{1054} \) using a calculator or estimation. The approximate value is:
\[ \sqrt{1054} \approx 32.5 \]

### Step 5: Combine the results
Now we can substitute back into our fraction:
\[ \sqrt{\frac{1054}{64}} = \frac{\sqrt{1054}}{8} \approx \frac{32.5}{8} \]

### Step 6: Simplify the fraction
Now we simplify \( \frac{32.5}{8} \):
\[ \frac{32.5}{8} = \frac{325}{80} = \frac{41}{10} \]

### Final Answer
Thus, the square root of \( \frac{1054}{64} \) is:
\[ \frac{41}{10} \text{ or } 4.1 \]