26. Выполните действия: 1–14.

Используем правила сокращения дробей и формулы разности квадратов.

1. \(\left(\frac{a}{2}-\frac{a}{3}\right)\cdot\frac{1}{a^2}=\frac{a}{6}\cdot\frac{1}{a^2}=\frac{1}{6a}\).
2. \(\frac{a^2}{3}\left(\frac{2}{a}+\frac{2}{a^2}\right)=\frac{2a}{3}+\frac{2}{3}=\frac{2(a+1)}{3}\).
3. \(\frac{a+b}{a-b}\left(\frac{a}{5}-\frac{b}{5}\right)=\frac{a+b}{a-b}\cdot\frac{a-b}{5}=\frac{a+b}{5}\).
4. \(\frac{ab}{a-b}\left(\frac{1}{b}-\frac{1}{a}\right)=\frac{ab}{a-b}\cdot\frac{a-b}{ab}=1\).
5. \(1:\left(1+\frac{1}{a}\right)=1:\frac{a+1}{a}=\frac{a}{a+1}\).
6. \(b:\left(b+\frac{1}{2}\right)=b:\frac{2b+1}{2}=\frac{2b}{2b+1}\).
7. \(\left(1+\frac{1}{a}\right):\left(1-\frac{1}{a}\right)=\frac{a+1}{a}:\frac{a-1}{a}=\frac{a+1}{a-1}\).
8. \(\left(a+\frac{a}{b}\right)\left(a-\frac{a}{b}\right)=a^2\left(1+\frac{1}{b}\right)\left(1-\frac{1}{b}\right)=\frac{a^2(b^2-1)}{b^2}\).
9. \(\left(\frac{b}{a}+\frac{a}{b}-2\right):\left(\frac{1}{b}-\frac{1}{a}\right)=\frac{(a-b)^2}{ab}:\frac{a-b}{ab}=a-b\).
10. \(\left(\frac{m}{n}+\frac{n}{m}+2\right)\left(1+\frac{m-n}{m+n}\right)=\frac{(m+n)^2}{mn}\cdot\frac{2m}{m+n}=\frac{2(m+n)}{n}\).
11. \(\left(1-\frac{a-b}{a+b}\right)\left(2+\frac{2b}{a-b}\right)=\frac{2b}{a+b}\cdot\frac{2a}{a-b}=\frac{4ab}{(a+b)(a-b)}\).
12. \(\left(1+\frac{a+b}{a-b}\right)\left(2-\frac{2a}{a+b}\right)=\frac{2a}{a-b}\cdot\frac{2b}{a+b}=\frac{4ab}{(a-b)(a+b)}\).
13. \(\left(\frac{6}{a-b}-\frac{5}{a+b}\right)\cdot\frac{a-b}{a+11b}=\frac{6(a+b)-5(a-b)}{(a-b)(a+b)}\cdot\frac{a-b}{a+11b}=\frac{a+11b}{(a+b)(a+11b)}=\frac{1}{a+b}\).
14. \(\left(\frac{3}{c}+\frac{3}{c+d}\right)\cdot\frac{c}{18(2c+d)}=\frac{3(2c+d)}{c(c+d)}\cdot\frac{c}{18(2c+d)}=\frac{1}{6(c+d)}\).

Ответ: 1) \(\frac{1}{6a}\); 2) \(\frac{2(a+1)}{3}\); 3) \(\frac{a+b}{5}\); 4) \(1\); 5) \(\frac{a}{a+1}\); 6) \(\frac{2b}{2b+1}\); 7) \(\frac{a+1}{a-1}\); 8) \(\frac{a^2(b^2-1)}{b^2}\); 9) \(a-b\); 10) \(\frac{2(m+n)}{n}\); 11) \(\frac{4ab}{(a+b)(a-b)}\); 12) \(\frac{4ab}{(a-b)(a+b)}\); 13) \(\frac{1}{a+b}\); 14) \(\frac{1}{6(c+d)}\).