`sqrt(110 (1)/(4))` is equal to

Options

A. `12.0`
B. 11.5
C. `11.0`
D. 10.5

Answer

To simplify the expression \( \sqrt{110 \frac{1}{4}} \), we will follow these steps:

### Step 1: Convert the mixed number to an improper fraction
The mixed number \( 1 \frac{1}{4} \) can be converted to an improper fraction. To do this, we multiply the whole number (1) by the denominator (4) and then add the numerator (1).

\[ 1 \frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{4 + 1}{4} = \frac{5}{4} \]

### Step 2: Multiply by 110
Now, we need to multiply 110 by the improper fraction we just found:

\[ 110 \cdot \frac{5}{4} = \frac{110 \times 5}{4} = \frac{550}{4} \]

### Step 3: Simplify the fraction
Next, we simplify \( \frac{550}{4} \):

\[ \frac{550 \div 2}{4 \div 2} = \frac{275}{2} \]

### Step 4: Take the square root
Now we need to find the square root of \( \frac{275}{2} \):

\[ \sqrt{\frac{275}{2}} = \frac{\sqrt{275}}{\sqrt{2}} \]

### Step 5: Simplify \( \sqrt{275} \)
We can simplify \( \sqrt{275} \):

\[ 275 = 25 \times 11 \quad \text{(since 25 is a perfect square)} \]
\[ \sqrt{275} = \sqrt{25 \times 11} = \sqrt{25} \times \sqrt{11} = 5\sqrt{11} \]

### Step 6: Combine the results
Now we substitute back into the expression:

\[ \sqrt{\frac{275}{2}} = \frac{5\sqrt{11}}{\sqrt{2}} \]

### Step 7: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{2} \):

\[ \frac{5\sqrt{11} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{5\sqrt{22}}{2} \]

### Final Answer
Thus, the simplified form of \( \sqrt{110 \frac{1}{4}} \) is:

\[ \frac{5\sqrt{22}}{2} \]

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