5/39.(39/.5/3+25/3.39/5)+(-30/7.7/11+7/11.16/3).5-17/3-(268/15-24/7)-(42/15-4)-(145/69.36/29):5/23+25.0,375.(-10).1

Let's break down and solve this expression step by step:

Given expression:

\[
\frac{5}{39} \cdot \left(\frac{39}{0.5 \cdot 3} + \frac{25}{3} \cdot \frac{39}{5}\right) + \left(\frac{-30}{7} \cdot \frac{7}{11} + \frac{7}{11} \cdot \frac{16}{3}\right) \cdot 0.5 - \frac{17}{3} - \left(\frac{268}{15} - \frac{24}{7}\right) - \left(\frac{42}{15} - 4\right) - \left(\frac{145}{69} \cdot \frac{36}{29}\right):\frac{5}{23} + 25 \cdot 0.375 \cdot (-10) \cdot 0.1
\]

### Step 1: Simplify the first term inside the parentheses

\[
\frac{39}{0.5 \cdot 3} + \frac{25}{3} \cdot \frac{39}{5}
\]

1. \( \frac{39}{0.5 \cdot 3} = \frac{39}{1.5} = 26 \)
2. \( \frac{25}{3} \cdot \frac{39}{5} = \frac{25 \cdot 39}{3 \cdot 5} = \frac{975}{15} = 65 \)

So,

\[
26 + 65 = 91
\]

Thus, the first term becomes:

\[
\frac{5}{39} \cdot 91 = \frac{5 \cdot 91}{39} = \frac{455}{39} \approx 11.67
\]

### Step 2: Simplify the second term inside the parentheses

\[
\frac{-30}{7} \cdot \frac{7}{11} + \frac{7}{11} \cdot \frac{16}{3}
\]

1. \( \frac{-30}{7} \cdot \frac{7}{11} = \frac{-210}{77} = -\frac{210}{77} = -\frac{30}{11} \approx -2.727 \)
2. \( \frac{7}{11} \cdot \frac{16}{3} = \frac{112}{33} \approx 3.394 \)

So,

\[
-\frac{30}{11} + \frac{112}{33} \approx -2.727 + 3.394 = 0.667
\]

Thus, the second term becomes:

\[
0.667 \cdot 0.5 = 0.334
\]

### Step 3: Simplify the next individual terms

1. \( \frac{17}{3} \approx 5.667 \)
2. \( \frac{268}{15} - \frac{24}{7} \approx 17.867 - 3.429 = 14.438 \)
3. \( \frac{42}{15} - 4 \approx 2.8 - 4 = -1.2 \)
4. \( \frac{145}{69} \cdot \frac{36}{29} \approx 2.101 \cdot 1.241 = 2.607 \)

### Step 4: Simplify the entire expression step by step

1. The term \( \frac{145}{69} \cdot \frac{36}{29} \) divided by \( \frac{5}{23} \):

\[
\frac{2.607}{\frac{5}{23}} = \frac{2.607 \cdot 23}{5} \approx \frac{59.961}{5} \approx 11.992
\]

2. Add everything together:

\[
11.67 + 0.334 - 5.667 - 14.438 - (-1.2) - 11.992 + (-10 \cdot 25 \cdot 0.375 \cdot 0.1)
\]

Simplifying the multiplication:

\[
-10 \cdot 25 \cdot 0.375 \cdot 0.1 = -93.75
\]

So, finally:

\[
11.67 + 0.334 - 5.667 - 14.438 + 1.2 - 11.992 - 93.75 \approx -112.643
\]

The final answer is approximately \(-112.643\).

Would you like more details or have any questions?

---

Here are some questions for further exploration:

1. What would happen if the coefficients in the expression were changed?
2. How would this expression change if it included logarithms?
3. Can you graph the function represented by a similar equation?
4. How does the result change with small modifications in the constants?
5. How can you apply this type of arithmetic in real-world problems?
6. What are the advantages of breaking down complex expressions step by step?
7. Could there be a different method to simplify this expression?
8. How does rounding at each step affect the final result?

**Tip:** Always perform operations in parentheses first to avoid errors in complex expressions.

Learn how to solve a complex arithmetic expression step by step with detailed explanations. This problem involves arithmetic operations with fractions and multiple parentheses. Suitable for advanced high school or college-level students.

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