$a)x-a\ne 0\wedge a+x\ne 0\wedge a\ne 0\wedge x\ne 0$  

$x\ne a\wedge x\ne -a\wedge a\ne 0\wedge x\ne 0$  

 

$\left(\frac{x}{x-a}-\frac{a}{a+x}\right):\left(\frac{x+a}{a}-\frac{x-a}{x}\right)=\left(\frac{x\left(x+a\right)}{\left(x-a\right)\left(x+a\right)}-\frac{a\left(x-a\right)}{\left(x+a\right)\left(x-a\right)}\right):\left(\frac{x\left(x+a\right)}{xa}-\frac{a\left(x-a\right)}{xa}\right)=$  

$=\left(\frac{x^2+ax-ax-a^2}{x^2-a^2}\right):\left(\frac{x^2+ax-ax-a^2}{xa}\right)$ $=\left(\frac{x^2+ax-ax-a^2}{x^2-a^2}\right):\left(\frac{x^2+ax-ax-a^2}{xa}\right)$  

$\left(\frac{x^2-a^2}{x^2-a^2}\right)\cdot \left(x\frac{a}{x^2-a^2}\right)=\frac{xa}{x^2-a^2}$  

 

$b)x\ne 0\wedge a^3-x^3\ne 0$  

$x\ne 0\wedge a\ne x$  

 

$\left(1+\frac{a}{x}+\frac{a}{x^2}\right)\left(1-\frac{a}{x}\right)\cdot \left(\frac{x^3}{a^3-x^3}\right)=\left(\frac{x^2}{x^2}+\frac{ax}{x^2}+\frac{a}{x^2}\right)\left(\frac{x}{x}-\frac{a}{x}\right)\cdot \left(\frac{x^3}{a^3-x^3}\right)=\left(\frac{x^2+ax+a}{x^2}\right)\left(\frac{x-a}{x}\right)\cdot \frac{x^3}{a^3-x^3}=$

$\frac{x^3-a^3}{x^3}\cdot \frac{x^3}{a^3-x^3}=1$  

 

$c)a\ne 0\wedge b\ne 0\wedge a+b\ne 0\wedge 1+a\ne 0$  

$a\ne 0\wedge b\ne 0\wedge a\ne -b\wedge a\ne -1$  

 

$\left[\left(\frac{a^2}{ab}-\frac{b^2}{ab}\right)\cdot \frac{1}{a+b}+a\left(\frac{b}{ab}-\frac{a}{ab}\right)\right]\cdot \frac{b}{1+a}=\left[\frac{a^2-b^2}{ab}\cdot \frac{1}{a+b}+a\left(\frac{b-a}{ab}\right)\right]\cdot \frac{b}{1+a}=$  

$\left[\frac{\left(a-b\right)\left(a+b\right)}{ab}\cdot \frac{1}{a+b}+\frac{ab-a^2}{ab}\right]\cdot \frac{b}{1+a}=\left[\frac{a-b}{ab}+\frac{ab-a^2}{ab}\right]\cdot \frac{b}{1+a}=\left[\frac{a-b+ab-a^2}{ab}\right]\cdot \frac{b}{1+a}=$  

$\frac{\left(a-b\right)-a\left(a-b\right)}{a}\cdot \frac{1}{1+a}=\frac{\left(a-b\right)\left(1-a\right)}{a\left(1+a\right)}$  

 

$d)x\ne 0\wedge y\ne 0\wedge \frac{1}{x}-\frac{1}{y}\ne 0$  

$x\ne 0\wedge y\ne 0\wedge x\ne y$  

 

$\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}=\frac{\frac{y}{xy}+\frac{x}{xy}}{\frac{y}{xy}-\frac{x}{xy}}=\frac{x+y}{xy}\cdot \frac{xy}{y-x}=\frac{x+y}{y-x}$