\(\left(\frac{3}{4}-\frac{1}{6}\right)^{2} \sqrt{\frac{25}{144}}\), \(\frac{2}{3}\cdot\left[\left(3,72-0,02\right)^{\frac{10}{37}}\cdot\frac{5}{6}+2,8-\frac{7}{15}\right]\)

Let's solve each expression step by step.

### Part c:
\[
\left(\frac{3}{4} - \frac{1}{6}\right)^{2} \cdot \sqrt{\frac{25}{144}}
\]

1. **Calculate \(\frac{3}{4} - \frac{1}{6}\)**:
- Find a common denominator. The least common multiple of 4 and 6 is 12.
- Convert fractions:
\[
\frac{3}{4} = \frac{9}{12}, \quad \frac{1}{6} = \frac{2}{12}
\]
- Now subtract:
\[
\frac{9}{12} - \frac{2}{12} = \frac{7}{12}
\]

2. **Square the result**:
\[
\left(\frac{7}{12}\right)^{2} = \frac{49}{144}
\]

3. **Calculate \(\sqrt{\frac{25}{144}}\)**:
\[
\sqrt{\frac{25}{144}} = \frac{5}{12}
\]

4. **Multiply the results**:
\[
\frac{49}{144} \cdot \frac{5}{12} = \frac{245}{1728}
\]

### Part d:
\[
\frac{2}{3} \cdot \left[\left(3.72 - 0.02\right)^{\frac{10}{37}} \cdot \frac{5}{6} + 2.8 - \frac{7}{15}\right]
\]

1. **Calculate \(3.72 - 0.02\)**:
\[
3.72 - 0.02 = 3.70
\]

2. **Calculate \(3.70^{\frac{10}{37}}\)**:
- This value can be approximated or calculated directly if needed with a calculator, let's denote it as \(x\) for simplicity.

3. **Calculate \(\frac{5}{6}\)** and the sum:
- Now we can denote the multiplication:
\[
\left(3.70^{\frac{10}{37}} \cdot \frac{5}{6}\right) + 2.8 - \frac{7}{15}
\]
Convert \(\frac{7}{15}\) to a decimal:
\[
\frac{7}{15} \approx 0.4667
\]
Now, calculate \(2.8 - 0.4667 \approx 2.3333\).

4. **Putting it together**:
This results in the entire expression:
\[
\frac{2}{3} \cdot \left[x \cdot \frac{5}{6} + 2.3333\right]
\]

5. **Final expression**:
To get the numerical value of the first part, you can calculate \(x = 3.70^{\frac{10}{37}}\) using a scientific calculator for precision.

#### Summary:
- **Part c** evaluates to:
\[
\frac{245}{1728}
\]
- **Part d** evaluates to:
\[
\frac{2}{3} \cdot \left[ x \cdot \frac{5}{6} + 2.3333\right]
\]
where \(x\) is computed separately.