Complex fractions

Example 1: Simplify $\dfrac{\dfrac{4x^2y^5}{5uv^3}}{\dfrac{5x^3y^4}{11u^3v^2}}.$

Solution: In this complex rational expression, there is one rational expression in the numerator and another in the denominator. First, we can write the division with the $\div$ symbol:

$\dfrac{\dfrac{4x^2y^5}{5uv^3}}{\dfrac{5x^3y^4}{11u^3v^2}}=\dfrac{4x^2y^5}{5uv^3} \div \dfrac{5x^3y^4}{11u^3v^2}$

Now, we can replace the $\div$ symbol with the multiplication symbol by taking the reciprocal of the divisor.

$\hphantom{00000}=\dfrac{4x^2y^5}{5uv^3} \cdot \dfrac{11u^3v^2}{5x^3y^4}$

Simplifying,

$\hphantom{00000}=\dfrac{44u^2y}{25vx}$

Example 2: Simplify, $\dfrac{\dfrac{u+5}{7}}{\dfrac{u^2-25}{u+2}}$

Solution:

$\dfrac{\dfrac{u+5}{7}}{\dfrac{u^2-25}{u+2}}=\dfrac{u+5}{7}\div \dfrac{u^2-25}{u+2}$

Take the reciprocal of the divisor and change division to multiplication.

$\hphantom{00000}=\dfrac{u+5}{7}\cdot \dfrac{u+2}{u^2-25}$

We need to factor $u^2-25$; that is, $u^2-25=(u+5)(u-5)$. Substituting this,

$\hphantom{00000}=\dfrac{u+5}{7}\cdot \dfrac{u+2}{(u+5)(u-5)}$

Canceling $u+5$,

$\hphantom{00000}=\dfrac{u+2}{7(u-5)}$

Example 3: Simplify, $\dfrac{6+\dfrac{15}{x}}{\dfrac{2}{x}+\dfrac{5}{x^2}}$

Solution:

The LCD of the fractions in the numerator is $x$, and the LCD of the fractions in the denominator is $x^2$. With these LCDs, we can write

$\dfrac{6+\dfrac{15}{x}}{\dfrac{2}{x}+\dfrac{5}{x^2}}=\dfrac{\dfrac{6x+15}{x}}{\dfrac{2x+5}{x^2}}$

Changing the division to multiplication by taking the reciprocal,

$\hphantom{00000}=\dfrac{6x+15}{x}\cdot \dfrac{x^2}{2x+5}$

Factoring $6x+15$,

$\phantom{00000}=\dfrac{3(2x+5)}{x}\cdot \dfrac{x^2}{2x+5}$

Canceling an $x$ and $2x+5$, we get

$\hphantom{00000}=\boxed{3x}$

Example 4: Simplify, $\dfrac{7v^{-1}y^{-1}}{2v^{-2}+5y^{-1}}$

Solution:

Making the exponents in the denominator positive,

$\dfrac{7v^{-1}y^{-1}}{2v^{-2}+5y^{-1}}=\dfrac{7v^{-1}y^{-1}}{\dfrac{2}{v^2}+\dfrac{5}{y}}$

The LCD of the fractions in the denominator is $v^2y$, so we can write

$\hphantom{00000}=\dfrac{7v^{-1}y^{-1}}{\dfrac{2y+5v^2}{v^2y}}$

$\hphantom{00000}=7v^{-1}y^{-1}\cdot \dfrac{v^2y}{2y+5v^2}$

$\hphantom{00000}=\boxed{\dfrac{7v}{2y+5v^2}}$