Transformation of Rational Expressions

Transformation of any rational expression reduces to addition, subtraction, multiplication and division, raising to natural power of rational fractions. We can transform any rational expression into fraction with the numerator and denominator, that are integer rational expressions; this is sense of identical transformation of rational expression.

Example. Simplify the following expression: `((2a)/(2a+2b)-(4a^2)/(4a^2+4ab+b^2))*((2a)/(4a^2-b^2)+1/(b-2a))^(-1)+(8a^2)/(2a+b)`.

  1. `(2a)/(2a+b)-(4a^2)/(4a^2+4ab+b^2)=(2a*color(red)((2a+b)))/(color(red)((2a+b)))-(4a^2)/(2a+b)^2=(2a(2a+b)-4a^2)/(2a+b)^2=(2ab)/(2a+b)^2`;
  2. `(2a)/(4a^2-b^2)+1/(b-2a)=(2a)/((2a-b)(2a+b))-(1*color(blue)((2a+b)))/((2a-b)color(blue)((2a+b)))=(2a-2a-b)/((2a-b)(2a+b))=(-b)/((2a-b)(2a+b))`;
  3. `(-b/((2a-b)(2a+b)))^(-1)=-((2a-b)(2a+b))/b`;
  4. `(2ab)/((2a+b)^2)*(-((2a-b)(2a+b))/b)=-(2ab(2a-b)(2a+b))/(b(2a+b)^2)=(2a(b-2a))/(2a+b)=(2ab-4a^2)/(2a+b)`;

`(2ab-4a^2)/(2a+b)+(8a^2)/(2a+b)=(2ab+4a^2)/(2a+b)=(2a(2a+b))/(2a+b)=2a`