Answered By Leonardo F

For adding and subtracting rational expressions, we have to apply the technique of leaving all of the expressions with the same denominator (common denominator), just like we do when adding or subtracting fractions. Examples: when adding:

 $\frac{x}{x+1}+\frac{1}{x}=\frac{x^2}{\left(x+1\right)x}+\frac{x+1}{\left(x+1\right)x}=\frac{x^2+x+1}{\left(x+1\right)x}$xx+1 +1x =x²(x+1)x +x+1(x+1)x =x²+x+1(x+1)x  

For rational numbers:

 $\frac{5}{7}+\frac{6}{10}=\frac{50}{70}+\frac{42}{70}=\frac{92}{70}=\frac{46}{35}$57 +610 =5070 +4270 =9270 =4635  

When subtracting:

 $\frac{x^2}{x+1}-\frac{1}{2x}=\frac{2x^3}{2x\left(x+1\right)}-\frac{x+1}{2x\left(x+1\right)}=\frac{2x^3-x-1}{2x\left(x+1\right)}$x²x+1 −12x =2x³2x(x+1) −x+12x(x+1) =2x³−x−12x(x+1)  

For rational numbers:

 $\frac{5}{6}-\frac{8}{7}=\frac{35}{42}-\frac{48}{42}=\frac{-13}{42}$56 −87 =3542 −4842 =−1342  

When multiplying or dividing, we don't have the restriction of the common denominator. For multiplication, we just have to multiply the numerators and the denominators. Example:

 $\left(\frac{x}{x+1}\right)\left(\frac{x^2}{2}\right)=\frac{x^3}{2\left(x+1\right)}$(xx+1 )(x²2 )=x³2(x+1)  

For rational numbers:

 $\left(\frac{5}{7}\right)\left(\frac{4}{13}\right)=\frac{20}{91}$(57 )(413 )=2091  

When dividing rational expressions, we can always transform the division into a multiplication by simply flipping the second expression. Example:

  $\frac{\frac{x}{x+1}}{\frac{x^2}{2x}}=\left(\frac{x}{x+1}\right)\left(\frac{2x}{x^2}\right)=\frac{2x^2}{x^2\left(x+1\right)}=\frac{2}{x+1}$xx+1 x²2x  =(xx+1 )(2xx² )=2x²x²(x+1) =2x+1   

For rational numbers:

 $\frac{\frac{5}{12}}{\frac{5}{3}}=\left(\frac{5}{12}\right)\left(\frac{3}{5}\right)=\frac{15}{60}=\frac{1}{4}$512 53  =(512 )(35 )=1560 =14