`sqrt((4 (1)/(7) - 2 (1)/(4))/(3 (1)/(2) + 1 (1)/(7)) + (2)/(2 + (1)/(2 + (1)/(5 - (1)/(5)))))` is equal to

Options

A. 1
B. 4
C. 3
D. 2

Answer

To solve the expression

\[ \sqrt{\frac{4 \frac{1}{7} - 2 \frac{1}{4}}{3 \frac{1}{2} + 1 \frac{1}{7}} + \frac{2}{2 + \frac{1}{2 + \frac{1}{5 - \frac{1}{5}}}}} \]

we will break it down step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers to improper fractions:

- \(4 \frac{1}{7} = \frac{4 \times 7 + 1}{7} = \frac{29}{7}\)
- \(2 \frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}\)
- \(3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}\)
- \(1 \frac{1}{7} = \frac{1 \times 7 + 1}{7} = \frac{8}{7}\)

### Step 2: Substitute the Improper Fractions
Now substitute these values back into the expression:

\[ \sqrt{\frac{\frac{29}{7} - \frac{9}{4}}{\frac{7}{2} + \frac{8}{7}} + \frac{2}{2 + \frac{1}{2 + \frac{1}{5 - \frac{1}{5}}}}} \]

### Step 3: Solve the Numerator and Denominator Separately
#### Numerator:
\[ \frac{29}{7} - \frac{9}{4} \]
To subtract these fractions, we need a common denominator. The least common multiple of 7 and 4 is 28.

\[ = \frac{29 \times 4}{28} - \frac{9 \times 7}{28} = \frac{116}{28} - \frac{63}{28} = \frac{53}{28} \]

#### Denominator:
\[ \frac{7}{2} + \frac{8}{7} \]
Again, we find a common denominator, which is 14.

\[ = \frac{7 \times 7}{14} + \frac{8 \times 2}{14} = \frac{49}{14} + \frac{16}{14} = \frac{65}{14} \]

### Step 4: Substitute Back into the Expression
Now substitute back into the expression:

\[ \sqrt{\frac{\frac{53}{28}}{\frac{65}{14}} + \frac{2}{2 + \frac{1}{2 + 0}}} \]

### Step 5: Simplify the Fraction
To simplify \(\frac{\frac{53}{28}}{\frac{65}{14}}\):

\[ = \frac{53}{28} \times \frac{14}{65} = \frac{53 \times 14}{28 \times 65} = \frac{53 \times 1}{2 \times 65} = \frac{53}{130} \]

### Step 6: Simplify the Second Term
Now simplify the second term:

\[ 2 + \frac{1}{2 + 0} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \]

So,

\[ \frac{2}{\frac{5}{2}} = 2 \times \frac{2}{5} = \frac{4}{5} \]

### Step 7: Combine the Two Parts
Now combine both parts:

\[ \sqrt{\frac{53}{130} + \frac{4}{5}} \]

Convert \(\frac{4}{5}\) to have a common denominator of 130:

\[ \frac{4}{5} = \frac{4 \times 26}{5 \times 26} = \frac{104}{130} \]

Now combine:

\[ \frac{53}{130} + \frac{104}{130} = \frac{157}{130} \]

### Step 8: Final Expression
Now we have:

\[ \sqrt{\frac{157}{130}} \]

### Step 9: Simplify the Square Root
This can be simplified to:

\[ \frac{\sqrt{157}}{\sqrt{130}} \]

### Conclusion
Thus, the final answer is:

\[ \sqrt{\frac{157}{130}} \]