Evaluate: `((15)/(243)xx(-27)/(75))+((8)/(-66)xx(-22)/(40))`

Answer

To evaluate the expression \(\left(\frac{15}{243} \times \frac{-27}{75}\right) + \left(\frac{8}{-66} \times \frac{-22}{40}\right)\), we will follow these steps:

### Step 1: Simplify the first term \(\frac{15}{243} \times \frac{-27}{75}\)

1. **Factor and simplify**:
- The fraction \(\frac{15}{75}\) simplifies to \(\frac{1}{5}\) because \(15 \div 15 = 1\) and \(75 \div 15 = 5\).
- The fraction \(\frac{-27}{243}\) simplifies to \(\frac{-1}{9}\) because \(-27 \div -27 = -1\) and \(243 \div -27 = 9\).

2. **Combine the simplified fractions**:
\[ \frac{15}{243} \times \frac{-27}{75} = \frac{1}{5} \times \frac{-1}{9} = \frac{-1}{45} \]

### Step 2: Simplify the second term \(\frac{8}{-66} \times \frac{-22}{40}\)

1. **Factor and simplify**:
- The fraction \(\frac{8}{-66}\) simplifies to \(\frac{-4}{33}\) because \(8 \div 2 = 4\) and \(-66 \div 2 = -33\).
- The fraction \(\frac{-22}{40}\) simplifies to \(\frac{-11}{20}\) because \(-22 \div 2 = -11\) and \(40 \div 2 = 20\).

2. **Combine the simplified fractions**:
\[ \frac{-4}{33} \times \frac{-11}{20} = \frac{44}{660} \]

3. **Further simplify \(\frac{44}{660}\)**:
- The GCD of 44 and 660 is 44, so:
\[ \frac{44 \div 44}{660 \div 44} = \frac{1}{15} \]

### Step 3: Combine the two simplified terms

Now we have:
\[ \frac{-1}{45} + \frac{1}{15} \]

1. **Find a common denominator**:
- The LCM of 45 and 15 is 45.

2. **Convert \(\frac{1}{15}\) to have a denominator of 45**:
\[ \frac{1}{15} = \frac{3}{45} \]

3. **Combine the fractions**:
\[ \frac{-1}{45} + \frac{3}{45} = \frac{-1 + 3}{45} = \frac{2}{45} \]

### Final Answer
Therefore, the final result of the expression is:
\[ \frac{2}{45} \]