The square root of `32 137/484` is :

Options

A. `5 15/22`
B. `15 5/32`
C. `5 15/28`
D. None of these

Answer

To find the square root of the mixed fraction \( 32 \frac{137}{484} \), we can follow these steps:

### Step 1: Convert the mixed fraction to an improper fraction
First, we convert the mixed fraction \( 32 \frac{137}{484} \) into an improper fraction.

\[ 32 \frac{137}{484} = \frac{32 \times 484 + 137}{484} \]

Calculating \( 32 \times 484 \):

\[ 32 \times 484 = 15488 \]

Now, adding 137:

\[ 15488 + 137 = 15625 \]

Thus, the improper fraction is:

\[ \frac{15625}{484} \]

### Step 2: Find the square root of the improper fraction
Now, we need to find the square root of \( \frac{15625}{484} \).

Using the property of square roots:

\[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]

So we have:

\[ \sqrt{\frac{15625}{484}} = \frac{\sqrt{15625}}{\sqrt{484}} \]

### Step 3: Calculate \( \sqrt{15625} \)
To find \( \sqrt{15625} \), we can use the division method or factorization.

After performing the division method or by recognizing that \( 15625 = 125^2 \):

\[ \sqrt{15625} = 125 \]

### Step 4: Calculate \( \sqrt{484} \)
Next, we calculate \( \sqrt{484} \).

Recognizing that \( 484 = 22^2 \):

\[ \sqrt{484} = 22 \]

### Step 5: Combine the results
Now we can combine the results:

\[ \sqrt{32 \frac{137}{484}} = \frac{\sqrt{15625}}{\sqrt{484}} = \frac{125}{22} \]

### Step 6: Convert to a mixed fraction (if needed)
To express \( \frac{125}{22} \) as a mixed fraction, we perform the division:

\[ 125 \div 22 = 5 \quad \text{(whole number)} \]
\[ 125 - (22 \times 5) = 125 - 110 = 15 \quad \text{(remainder)} \]

Thus,

\[ \frac{125}{22} = 5 \frac{15}{22} \]

### Final Answer
The square root of \( 32 \frac{137}{484} \) is:

\[ 5 \frac{15}{22} \]

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