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The expression given is:

\[
\left( \frac{5}{12} - \frac{7}{9} \cdot 0.25 \right)^{-1} \cdot 1 \cdot \frac{2}{9}
\]

Let's solve this step-by-step.

### Step 1: Evaluate the inner expression
First, calculate the term inside the parentheses:

\[
\frac{7}{9} \cdot 0.25 = \frac{7 \times 0.25}{9} = \frac{1.75}{9}
\]

Now subtract this from \( \frac{5}{12} \):

\[
\frac{5}{12} - \frac{1.75}{9}
\]

To subtract these fractions, we need a common denominator. The least common denominator between 12 and 9 is 36, so:

\[
\frac{5}{12} = \frac{15}{36}, \quad \frac{1.75}{9} = \frac{7}{36}
\]

Now subtract:

\[
\frac{15}{36} - \frac{7}{36} = \frac{8}{36} = \frac{2}{9}
\]

### Step 2: Apply the exponent
Now, apply the exponent of \( -1 \), which means taking the reciprocal:

\[
\left( \frac{2}{9} \right)^{-1} = \frac{9}{2}
\]

### Step 3: Multiply the remaining terms
Now multiply by 1 and \( \frac{2}{9} \):

\[
\frac{9}{2} \cdot 1 \cdot \frac{2}{9} = \frac{9}{2} \cdot \frac{2}{9}
\]

Simplify:

\[
\frac{9 \times 2}{2 \times 9} = 1
\]

### Final Answer:
The value of the expression is \( \boxed{1} \).

Would you like more details, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. What is the significance of finding a common denominator when subtracting fractions?
2. How does multiplying by a reciprocal affect an equation?
3. Can you provide another example involving fractional exponents?
4. Why do we take the reciprocal when applying a \( -1 \) exponent?
5. How would the answer change if any value in the fraction were different?

**Tip:** When dealing with fractions, always simplify intermediate steps to avoid errors in calculation!

Find the value of ((5/12 - 7/9 * 0.25)^-1 * 1 * 2/9). All answers must be given as integers (whole numbers) or decimal numbers with a decimal point.

Learn how to handle complex fraction operations combined with negative exponents. This example breaks down the steps to subtract and multiply fractions, apply negative exponents, and find the reciprocal. This problem is appropriate for middle school and high school students.

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