Simplify: `[ (5)/(23) + (- (18)/(115)) + ((- 28)/(138)) ]xx [ ((23)/(14)) div ((69)/(17)) ]`

Answer

To simplify the expression \[ \left( \frac{5}{23} + \left(-\frac{18}{115}\right) + \left(-\frac{28}{138}\right) \right) \times \left( \frac{23}{14} \div \frac{69}{17} \right) \], we will follow these steps:

### Step 1: Simplify the Division
First, we simplify the division part of the expression:
\[ \frac{23}{14} \div \frac{69}{17} = \frac{23}{14} \times \frac{17}{69} \]
This can be simplified to:
\[ \frac{23 \times 17}{14 \times 69} \]

### Step 2: Find the LCM of Denominators
Next, we will find the LCM of the denominators in the first part of the expression: \(23\), \(115\), and \(138\).

- The prime factorization of \(23\) is \(23\).
- The prime factorization of \(115\) is \(5 \times 23\).
- The prime factorization of \(138\) is \(2 \times 3 \times 23\).

The LCM is:
\[ 23 \times 5 \times 2 \times 3 = 690 \]

### Step 3: Rewrite Each Fraction with LCM
Now we will rewrite each fraction with the common denominator \(690\):
- For \(\frac{5}{23}\):
\[ \frac{5}{23} = \frac{5 \times 30}{23 \times 30} = \frac{150}{690} \]
- For \(-\frac{18}{115}\):
\[ -\frac{18}{115} = -\frac{18 \times 6}{115 \times 6} = -\frac{108}{690} \]
- For \(-\frac{28}{138}\):
\[ -\frac{28}{138} = -\frac{28 \times 5}{138 \times 5} = -\frac{140}{690} \]

### Step 4: Combine the Fractions
Now we can combine the fractions:
\[ \frac{150}{690} - \frac{108}{690} - \frac{140}{690} = \frac{150 - 108 - 140}{690} = \frac{-98}{690} \]

### Step 5: Simplify the Result
Now we will simplify the result:
\[ \frac{-98}{690} \]
The GCD of \(98\) and \(690\) is \(14\), so we can simplify:
\[ \frac{-98 \div 14}{690 \div 14} = \frac{-7}{49} \]

### Step 6: Multiply by the Simplified Division
Now we multiply by the simplified division:
\[ \frac{-7}{49} \times \frac{23 \times 17}{14 \times 69} \]
Calculating the multiplication:
\[ = \frac{-7 \times 23 \times 17}{49 \times 14 \times 69} \]
Since \(49 = 7 \times 7\), we can simplify:
\[ = \frac{-1 \times 23 \times 17}{7 \times 14 \times 69} \]

### Final Result
Calculating the final result gives:
\[ = \frac{-391}{7 \times 14 \times 69} \]