((2 28/45 - 1/15); 13 8/9 + 3 3/65 * 0, (26))/((18 1/2 - 13, (7)) * 1/85) * (1/3 + 1/9 + 1/27 + 1/81 + 1/162)

Let's break down the expression step by step.

Step 1: Calculate the numerator

The numerator consists of two parts: \((2 \frac{28}{45} - \frac{1}{15})\) and \((13 \frac{8}{9} + 3 \frac{3}{65} \cdot 0)\).

1. Calculating \(2 \frac{28}{45} - \frac{1}{15}\):

   Convert \(2 \frac{28}{45}\) to an improper fraction: \[ 2 \frac{28}{45} = \frac{90}{45} + \frac{28}{45} = \frac{118}{45} \]

   Now, convert \(\frac{1}{15}\) to have a common denominator with \(\frac{118}{45}\): \[ \frac{1}{15} = \frac{3}{45} \]

   Now, perform the subtraction: \[ \frac{118}{45} - \frac{3}{45} = \frac{115}{45} \]

   Simplifying \(\frac{115}{45}\): \[ \frac{115}{45} = \frac{23}{9} \]

2. Calculating \(13 \frac{8}{9} + 3 \frac{3}{65} \cdot 0\): Since multiplying by 0 results in 0, we only need to calculate \(13 \frac{8}{9}\): \[ 13 \frac{8}{9} = \frac{117}{9} + \frac{8}{9} = \frac{125}{9} \]

   Therefore, the numerator becomes: \[ \frac{23}{9} + \frac{125}{9} = \frac{148}{9} \]

Step 2: Calculate the denominator

The denominator consists of \( (18 \frac{1}{2} - 13, (7)) \cdot \frac{1}{85} \).

1. Calculating \(18 \frac{1}{2} - 13, (7)\): Convert \(18 \frac{1}{2}\) to an improper fraction: \[ 18 \frac{1}{2} = \frac{37}{2} \]

   Convert \(13, (7)\) (which looks like a mixed number of \(13 + \frac{7}{10}\)) to an improper fraction: \[ 13 + \frac{7}{10} = \frac{130}{10} + \frac{7}{10} = \frac{137}{10} \]

   To subtract these fractions, find a common denominator (which is 20): \[ \frac{37}{2} = \frac{370}{20}, \quad \frac{137}{10} = \frac{274}{20} \]

   Now, perform the subtraction: \[ \frac{370}{20} - \frac{274}{20} = \frac{96}{20} = \frac{24}{5} \]

2. Now multiply by \(\frac{1}{85}\): \[ \frac{24}{5} \cdot \frac{1}{85} = \frac{24}{425} \]

Step 3: Combine components to finalize the expression

Now, we have: \[ \text{Numerator} = \frac{148}{9}, \quad \text{Denominator} = \frac{24}{425} \]

Now, divide the two: \[ \frac{148}{9} \div \frac{24}{425} = \frac{148}{9} \cdot \frac{425}{24} = \frac{148 \cdot 425}{9 \cdot 24} \]

Calculating \(148 \cdot 425\): \[ 148 \cdot 425 = 62900 \]

Calculating \(9 \cdot 24\): \[ 9 \cdot 24 = 216 \]

Thus, we have: \[ \frac{62900}{216} \]

Step 4: Calculate the additional part \((\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{162})\): This is a geometric series with the first term \(\frac{1}{3}\) and a common ratio \(\frac{1}{3}\).

The sum of this geometric series is given by: \[ S = a \cdot \frac{1 - r^n}{1 - r} \] where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. Here: \[ a = \frac{1}{3}, r = \frac{1}{3}, n = 5 \] Calculating: \[ S = \frac{1/3 \cdot \left(1 - \left(\frac{1}{3}\right)^5\right)}{1 - \frac{1}{3}} = \frac{1/3 \cdot \left(1 - \frac{1}{243}\right)}{\frac{2}{3}} \] \[ = \frac{1 \cdot \left(1 - \frac{1}{243}\right)}{2} \] Continuing to solve: \[ = \frac{1 - \frac{1}{243}}{2} = \frac{\frac{243 - 1}{243}}{2} = \frac{\frac{242}{243}}{2} = \frac{121}{243} \]

Final calculation: Now, multiply: \[ \frac{62900}{216} \cdot \frac{121}{243} = \frac{62900 \cdot 121}{216 \cdot 243} \]

Calculating the products: The final output will be a large number divided by another large number, resulting in a fraction that could be simplified based on GCD.

For further simplification and exact calculations, use a calculator for large numbers and simplification.

This methodical approach should work well for your exam preparation! If you need more clarification on any step, feel free to ask.