Let's solve each part of the problem step by step.

### a) \(\frac{1}{7} - \frac{3}{8} + \frac{13}{7} \times \frac{1}{7}\)

First, calculate the multiplication:
\[
\frac{13}{7} \times \frac{1}{7} = \frac{13 \times 1}{7 \times 7} = \frac{13}{49}
\]

Next, find a common denominator for \(\frac{1}{7}\) and \(\frac{3}{8}\):
\[
\text{Common denominator} = 56
\]
\[
\frac{1}{7} = \frac{8}{56}, \quad \frac{3}{8} = \frac{21}{56}
\]
\[
\frac{8}{56} - \frac{21}{56} = \frac{8 - 21}{56} = \frac{-13}{56}
\]

Now, add \(\frac{-13}{56}\) and \(\frac{13}{49}\):
\[
\text{Common denominator} = 392
\]
\[
\frac{-13}{56} = \frac{-91}{392}, \quad \frac{13}{49} = \frac{104}{392}
\]
\[
\frac{-91}{392} + \frac{104}{392} = \frac{-91 + 104}{392} = \frac{13}{392}
\]

### b) \(\frac{1}{6} - \left(\frac{1}{6} - \frac{1}{4}\right)\)

First, calculate the subtraction inside the parentheses:
\[
\frac{1}{6} - \frac{1}{4} = \frac{2}{12} - \frac{3}{12} = \frac{-1}{12}
\]

Now, subtract:
\[
\frac{1}{6} - \left(\frac{-1}{12}\right) = \frac{1}{6} + \frac{1}{12} = \frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}
\]

### c) \(\frac{1}{2} - \frac{1}{3} + \frac{1}{23} + \frac{1}{6}\)

Find a common denominator for all fractions:
\[
\text{Common denominator} = 138
\]
\[
\frac{1}{2} = \frac{69}{138}, \quad \frac{1}{3} = \frac{46}{138}, \quad \frac{1}{23} = \frac{6}{138}, \quad \frac{1}{6} = \frac{23}{138}
\]
\[
\frac{69}{138} - \frac{46}{138} + \frac{6}{138} + \frac{23}{138} = \frac{69 - 46 + 6 + 23}{138} = \frac{52}{138} = \frac{26}{69} = \frac{2}{3}
\]

### d) \(\frac{5}{11} - \frac{3}{7} + \frac{5}{8} - \frac{8}{7} + 2 \frac{5}{7}\)

Convert the mixed number to an improper fraction:
\[
2 \frac{5}{7} = \frac{19}{7}
\]

Find a common denominator for all fractions:
\[
\text{Common denominator} = 616
\]
\[
\frac{5}{11} = \frac{280}{616}, \quad \frac{3}{7} = \frac{264}{616}, \quad \frac{5}{8} = \frac{385}{616}, \quad \frac{8}{7} = \frac{704}{616}, \quad \frac{19}{7} = \frac{1672}{616}
\]
\[
\frac{280}{616} - \frac{264}{616} + \frac{385}{616} - \frac{704}{616} + \frac{1672}{616} = \frac{280 - 264 + 385 - 704 + 1672}{616} = \frac{1369}{616} = \frac{1369}{616}
\]

### e) \(25 \cdot \left(-\frac{1}{5}\right)^3 + \frac{1}{5} - 2 \cdot \left(-\frac{1}{2}\right)^2 - \frac{1}{2}\)

Calculate each term:
\[
\left(-\frac{1}{5}\right)^3 = -\frac{1}{125}
\]
\[
25 \cdot -\frac{1}{125} = -\frac{25}{125} = -\frac{1}{5}
\]
\[
\left(-\frac{1}{2}\right)^2 = \frac{1}{4}
\]
\[
2 \cdot \frac{1}{4} = \frac{1}{2}
\]

Now, combine all terms:
\[
-\frac{1}{5} + \frac{1}{5} - \frac{1}{2} - \frac{1}{2} = 0 - 1 = -1
\]

### g) \(\frac{5^{12} \cdot 9^{0.5}}{3^{18} \cdot 2^{50}}\)

Simplify the numerator and denominator:
\[
9^{0.5} = 3
\]
\[
5^{12} \cdot 3
\]
\[
3^{18} = 3^{18}
\]

Combine:
\[
\frac{5^{12} \cdot 3}{3^{18} \cdot 2^{50}} = \frac{5^{12}}{3^{17} \cdot 2^{50}}
\]

### h) \(\frac{4^9 \cdot 6^6 \cdot 120}{8^4 \cdot 3^{12} \cdot 6^{11}}\)

Simplify each term:
\[
4^9 = (2^2)^9 = 2^{18}
\]
\[
6^6 = (2 \cdot 3)^6 = 2^6 \cdot 3^6
\]
\[
120 = 2^3 \cdot 3 \cdot 5
\]
\[
8^4 = (2^3)^4 = 2^{12}
\]
\[
3^{12} = 3^{12}
\]
\[
6^{11} = (2 \cdot 3)^{11} = 2^{11} \cdot 3^{11}
\]

Combine:
\[
\frac{2^{18} \cdot 2^6 \cdot 3^6 \cdot 2^3 \cdot 3 \cdot 5}{2^{12} \cdot 3^{12} \cdot 2^{11} \cdot 3^{11}} = \frac{2^{27} \cdot 3^7 \cdot 5}{2^{23} \cdot 3^{23}} = \frac{2^4 \cdot 5}{3^{16}}
\]

### k) \(\frac{4^2 \cdot 25^2 + 32.125}{2^5 \cdot 5^2}\)

Simplify each term:
\[
4^2 = 16
\]
\[
25^2 = 625
\]
\[
16 \cdot 625 = 10000
\]
\[
2^5 = 32
\]
\[
5^2 = 25
\]

Combine:
\[
\frac{10000 + 32.125}{32 \cdot 25} = \frac{10032.125}{800} = 12.54
\]

### m) \(\frac{5^7 \cdot 7^{12}}{7^5 \cdot 13 \cdot 7}\)

Simplify each term:
\[
5^7 = 5^7
\]
\[
7^{12} = 7^{12}
\]
\[
7^5 = 7^5
\]
\[
13 = 13
\]

Combine:
\[
\frac{5^7 \cdot 7^{12}}{7^5 \cdot 13 \cdot 7} = \frac{5^7 \cdot 7^7}{13} = \frac{5^7 \cdot 7^7}{13}
\]

These are the simplified results for each part of the problem.