Multiplying Rational Expressions Examples,
and Dividing

When you encounter multiplying rational expressions examples in Math, the general process conveniently follows the same approach as multiplying fractions that contain numbers.

Which we can look at a recap of.

Multiplying Number Fractions:

$\bf{\frac{2}{3} \space {\scriptsize{\times}} \space \frac{5}{7}}$    =    $\bf{\frac{2 \space \times \space 5}{3 \space \times \space 7}}$    =    $\bf{\frac{10}{21}}$

General case:

$\frac{a}{b} \space {\scriptsize{\times}} \space \frac{c}{d}$    =    $\frac{a \space \times \space c}{b \space \times \space d}$

Multiplying Rational Expressions:

If we have rational expressions    $\frac{Q(x)}{R(x)}$   and   $\frac{S(x)}{T(x)}$.


Then,    $\frac{Q(x)}{R(x)} \space {\scriptsize{\times}} \space \frac{S(x)}{T(x)}$   =   $\frac{Q(x) \space \times \space S(x)}{R(x) \space \times \space T(x)}$.


Example

$\frac{2}{3x} {\scriptsize{\times}} \frac{x^2}{5x}$   =   $\frac{2}{3x} {\scriptsize{\times}} \frac{x^2}{5}$   =   $\frac{2 \space \times \space x^2}{3x \space \times \space 5}$   =   $\frac{2x^2}{15x}$   =   $\frac{2x}{15}$

How to Multiply Rational Expressions:

1)  Look to factor the numerator and/or denominator if possible.
2)  Eliminate any common factors between the numerator and the denominator.
3)  Carry out any necessary multiplication and factor/simply further if possible.

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Multiplying Rational Expressions Examples

(1.1) 

3 × $\frac{x + 4}{9}$

Solution   

3 × $\frac{x + 4}{9}$   =   $\frac{3}{1} {\scriptsize{\times}} \frac{x + 4}{9}$   =   $\frac{3(x + 4)}{9}$   =   $\frac{x + 4}{3}$



(1.2) 

$\frac{5x^3}{4} {\scriptsize{\times}} \frac{3}{8x}$

Solution   

$\frac{5x^3}{4} {\scriptsize{\times}} \frac{3}{8x}$   =   $\frac{5x^3 \space \times \space 3}{4 \space \times \space 8x}$   =   $\frac{15x^3}{32x}$   =   $\frac{15x^3}{32}$



(1.3) 

$\frac{4a}{5a^2} {\scriptsize{\times}} \frac{10}{11a}$

Solution   

$\frac{4a}{5a^2} {\scriptsize{\times}} \frac{10}{11a}$   =   $\frac{4}{5a} {\scriptsize{\times}} \frac{10}{11a}$   =   $\frac{4 \space \times \space 10}{5a \space \times \space 11a}$   =   $\frac{40}{55a^2}$   =   $\frac{8}{11a^2}$



(1.4) 

$\frac{x^2 \space + \space 6x \space + \space 8}{3x^2 \space + \space x \space {\text{--}} \space 4} {\scriptsize{\times}} \frac{5x^2}{x \space + \space 4}$

Solution   

With multiplying rational expressions examples such as this.
We look to factor first where possible.

$\frac{x^2 \space + \space 6x \space + \space 8}{3x^2 \space + \space x \space {\text{--}} \space 4} {\scriptsize{\times}} \frac{5x^2}{x \space + \space 4}$   =   $\frac{(x \space + \space 2)(x \space + \space 4)}{(3x \space {\text{--}} \space 2)(x \space + \space 2)} {\scriptsize{\times}} \frac{5x^2}{x \space + \space 4}$

=   $\frac{(x \space + \space 4)}{(3x \space {\text{--}} \space 2)} {\scriptsize{\times}} \frac{5x^2}{x \space + \space 4}$   =   $\frac{5x^2(x \space + \space 4)}{(3x \space {\text{--}} \space2)(x \space + \space 4)}$   =   $\frac{5x^2}{3x \space {\text{--}} \space2}$



(1.5) 

$\frac{6a^2b}{5a} {\scriptsize{\times}} \frac{3b}{2a}$

Solution   

$\frac{6a^2b}{5a} {\scriptsize{\times}} \frac{3b}{2a}$   =   $\frac{18a^2b^2}{10a^2}$   =   $\frac{18b^2}{10}$   =   $\frac{9b^2}{5}$

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Dividing Rational Expressions

Dividing rational expressions follows almost the same method as we use in multiplying rational expressions examples.

Just like with dividing fractions involving whole numbers, we multiply once we flip one of the fractions present in the division sum.

If we had,     $\bf{\frac{1}{4} {\scriptsize{\div}} \frac{3}{7}}$.

To do this division sum we just perform the multiplication,     $\bf{\frac{1}{4} {\scriptsize{\times}} \frac{7}{3} \space {\scriptsize{=}} \space \frac{7}{12}}$.

We follow the same approach with rational expressions.



Example    


(2.1) 

$\frac{2x}{5} \space {\scriptsize{\div}} \space \frac{x \space {\text{--}} \space 2}{x \space + \space 1}$

Solution   

$\frac{2x}{5} \space {\scriptsize{\div}} \space \frac{x \space {\text{--}} \space 2}{x \space + \space 1}$   =   $\frac{2x}{5} {\scriptsize{\times}} \frac{x \space + \space 1}{x \space {\text{--}} \space 2}$   =   $\frac{2x(x \space + \space 1)}{5(x \space {\text{--}} \space 2)}$

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