Find the value of
`1/(4(2/7)) + 1/(3(11/13)) + 1/(5/9)`

Answer

To solve the expression \( \frac{1}{4 \left( \frac{2}{7} \right)} + \frac{1}{3 \left( \frac{11}{13} \right)} + \frac{1}{\frac{5}{9}} \), we will follow these steps:

### Step 1: Simplify each term
First, we need to simplify each term in the expression.

1. **First term:**
\[ \frac{1}{4 \left( \frac{2}{7} \right)} = \frac{1}{\frac{8}{7}} = \frac{7}{8} \]

2. **Second term:**
\[ \frac{1}{3 \left( \frac{11}{13} \right)} = \frac{1}{\frac{33}{13}} = \frac{13}{33} \]

3. **Third term:**
\[ \frac{1}{\frac{5}{9}} = \frac{9}{5} \]

### Step 2: Rewrite the expression
Now we can rewrite the expression with the simplified terms:
\[ \frac{7}{8} + \frac{13}{33} + \frac{9}{5} \]

### Step 3: Find the LCM of the denominators
The denominators are 8, 33, and 5. We need to find the least common multiple (LCM) of these numbers.

- The prime factorization of the denominators:
- \( 8 = 2^3 \)
- \( 33 = 3 \times 11 \)
- \( 5 = 5^1 \)

The LCM will be:
\[ LCM = 2^3 \times 3^1 \times 5^1 \times 11^1 = 8 \times 3 \times 5 \times 11 = 1320 \]

### Step 4: Convert each fraction to have the common denominator
Now we convert each fraction to have the denominator of 1320.

1. **Convert \( \frac{7}{8} \):**
\[ \frac{7}{8} = \frac{7 \times 165}{8 \times 165} = \frac{1155}{1320} \]

2. **Convert \( \frac{13}{33} \):**
\[ \frac{13}{33} = \frac{13 \times 40}{33 \times 40} = \frac{520}{1320} \]

3. **Convert \( \frac{9}{5} \):**
\[ \frac{9}{5} = \frac{9 \times 264}{5 \times 264} = \frac{2376}{1320} \]

### Step 5: Add the fractions
Now we can add the fractions together:
\[ \frac{1155}{1320} + \frac{520}{1320} + \frac{2376}{1320} = \frac{1155 + 520 + 2376}{1320} = \frac{4051}{1320} \]

### Step 6: Simplify the final fraction
The fraction \( \frac{4051}{1320} \) cannot be simplified further since 4051 is a prime number.

### Final Answer
Thus, the final answer is:
\[ \frac{4051}{1320} \]