Let's solve the expression step by step.

The expression is:

[
39 / \left(\frac{3}{8} + \frac{1}{6}\right) + \left(\frac{3}{10}\right)^2 \left(\frac{2}{3} - \frac{7}{18}\right)
]

Step 1: Calculate (\frac{3}{8} + \frac{1}{6})

To add (\frac{3}{8}) and (\frac{1}{6}), we need a common denominator. The least common multiple of 8 and 6 is 24.

Convert (\frac{3}{8}) to have a denominator of 24:
[
\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}
]Convert (\frac{1}{6}) to have a denominator of 24:
[
\frac{1}{6} = \frac{1 \times 4}{6 \times 4} = \frac{4}{24}
]
Now we can add the fractions:
[
\frac{3}{8} + \frac{1}{6} = \frac{9}{24} + \frac{4}{24} = \frac{13}{24}
]

Step 2: Calculate (39 / \left(\frac{3}{8} + \frac{1}{6}\right))

Now that we found (\frac{3}{8} + \frac{1}{6} = \frac{13}{24}), we compute:
[
39 / \left(\frac{13}{24}\right) = 39 \times \frac{24}{13}
]
Calculating this gives:
[
39 \times \frac{24}{13} = \frac{936}{13} = 72
]

Step 3: Calculate (\left(\frac{3}{10}\right)^2)

Next, we compute (\left(\frac{3}{10}\right)^2):
[
\left(\frac{3}{10}\right)^2 = \frac{9}{100}
]

Step 4: Calculate (\frac{2}{3} - \frac{7}{18})

To subtract (\frac{2}{3}) and (\frac{7}{18}), we need a common denominator. The least common multiple of 3 and 18 is 18.

Convert (\frac{2}{3}) to have a denominator of 18:
[
\frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18}
]
Now subtract:
[
\frac{2}{3} - \frac{7}{18} = \frac{12}{18} - \frac{7}{18} = \frac{5}{18}
]

Step 5: Combine ( \left(\frac{3}{10}\right)^2 \times \left(\frac{2}{3} - \frac{7}{18}\right) )

Thus, we have:
[
\frac{9}{100} \times \frac{5}{18} = \frac{9 \times 5}{100 \times 18} = \frac{45}{1800}
]
Simplifying (\frac{45}{1800}) gives:
[
\frac{45 \div 45}{1800 \div 45} = \frac{1}{40}
]

Step 6: Putting it all together

Now we can add the two parts:
[
72 + \frac{1}{40}
]

To add these, convert 72 to have a denominator of 40:
[
72 = \frac{72 \times 40}{1 \times 40} = \frac{2880}{40}
]
Adding:
[
\frac{2880}{40} + \frac{1}{40} = \frac{2880 + 1}{40} = \frac{2881}{40}
]

Thus, the final result is:

[
\boxed{\frac{2881}{40}}
] or (72.025) in decimal form.