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The mathematical expression in the image is somewhat complex and appears to involve fractions, exponents, and nested operations. Let me transcribe the expression and then solve it step by step.

The expression is:

\[
\frac{\left( \frac{1 \times 3}{5} - \frac{2}{3 \times 3} \right)^3}{\frac{\frac{3}{2 \times 3} \times x}{\frac{13}{9}}}
\]

Let me first simplify and solve this step by step.

### Step 1: Simplify the numerator
1. First, simplify the inner terms of the numerator:
   \[
   \frac{1 \times 3}{5} = \frac{3}{5}
   \]
   \[
   \frac{2}{3 \times 3} = \frac{2}{9}
   \]
   Now, subtract these two fractions:
   \[
   \frac{3}{5} - \frac{2}{9} = \frac{(3 \times 9) - (2 \times 5)}{5 \times 9} = \frac{27 - 10}{45} = \frac{17}{45}
   \]

2. Raise the result to the power of 3:
   \[
   \left( \frac{17}{45} \right)^3 = \frac{17^3}{45^3} = \frac{4913}{91125}
   \]

### Step 2: Simplify the denominator
1. First, simplify the fractions:
   \[
   \frac{3}{2 \times 3} = \frac{3}{6} = \frac{1}{2}
   \]
   So the expression becomes:
   \[
   \frac{\frac{1}{2} \times x}{\frac{13}{9}}
   \]
2. Invert and multiply the bottom part:
   \[
   \frac{1}{2} \times x \times \frac{9}{13} = \frac{9x}{26}
   \]

### Step 3: Put the simplified numerator and denominator together
Now the expression becomes:
\[
\frac{\frac{4913}{91125}}{\frac{9x}{26}}
\]
Simplify this complex fraction by multiplying by the reciprocal:
\[
\frac{4913}{91125} \times \frac{26}{9x} = \frac{4913 \times 26}{91125 \times 9x} = \frac{127738}{820125x}
\]

Thus, the final simplified result is:
\[
\frac{127738}{820125x}
\]

Would you like further clarification or details on any step?

Here are five related questions to explore:
1. How can you simplify more complex fractions like the one in this expression?
2. What are the common strategies for working with exponents in fractional form?
3. Can you explore the process of subtracting fractions with unlike denominators?
4. How can nested fractions be simplified systematically?
5. How does multiplying by the reciprocal work when dealing with complex fractions?

**Tip**: Always simplify the expression inside the parentheses or brackets first, especially in cases with nested fractions, before applying exponents or other operations.

Solve the complex mathematical fraction expression shown in the image.

Learn how to solve a complex fraction involving exponents, subtraction of fractions, and nested operations. This detailed step-by-step solution simplifies both the numerator and the denominator to arrive at the final result. Suitable for students in Grades 9-12.

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