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# Multiplying Rational Expressions

A rational expression is a polynomial term, in the form of fraction. It may be any type of polynomial such as monomial, trinomial and binomial. Multiplication of rational expression is same as normal multiplication of fractions. A rational expression has a polynomial in both numerator or only in denominator.

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## How to Multiply Rational Expressions?

Below are the steps to multiply rational expressions.

Step 1: First do the multiplication of numerators of the rational expressions. If possible, factor the obtained result.

Step 2: Do the multiplication of denominators of rational expression and if possible factor the resultant value.

Step 3: Now, write the result by writing fraction which has above two answers. The final answer is again simplified if possible.

## Multiplying and Simplifying Rational Expressions

Let us see an example on how to multiply and simplify rational expressions.

Let us multiply and simplify rational expressions $\frac{8x}{x^2 + 14x + 49}$ x $\frac{x^2 + 2x - 35}{12x^6}$

Given rational expressions are, $\frac{8x}{x^2 + 14x + 49}$ x $\frac{x^2 + 2x - 35}{12x^6}$

Step 1:

The factor of a first rational expression is,

$\frac{8x}{x^2 + 14x + 49}$ = $\frac{8x}{(x + 7)(x + 7)}$

Step 2:

The factors of second rational expression is,

$\frac{x^2 + 2x - 35}{12x^6}$ = $\frac{(x + 7)(x - 5)}{12x^6}$

Step 3:

Now, the multiplication of rational expressions is,

$\frac{8x}{(x + 7)(x + 7)}$ x $\frac{(x + 7)(x - 5)}{12x^6}$

The common factor in both rational expression is (x + 7) and x.

$\frac{8x}{(x + 7)(x + 7)}$ x $\frac{(x + 7)(x - 5)}{12x^6}$ = $\frac{8}{x + 7}$ x $\frac{(x - 5)}{12x^5}$

= $\frac{8(x - 5)}{(x + 7)(12x^5)}$

We can also cancel the numbers 8 and 12 by 4.

$\frac{8(x - 5)}{(x + 7)(12x^5)}$ = $\frac{2(x - 5)}{(x + 7)(3x^5)}$

## Multiplying Rational Expressions Examples

Below you could see some examples of multiplication of rational expression:

### Solved Examples

Question 1: Multiply $\frac{6x^5}{x^2+16x+63}$ and $\frac{x^2+4x-21}{9x^4}$
Solution:

Given expressions are $\frac{6x^5}{x^2+16x+63}$ and $\frac{x^2+4x-21}{9x^4}$

Step 1:
Simplify first expression, $\frac{6x^5}{x^2+16x+63}$

= $\frac{6x^5}{(x + 9)(x + 7)}$

Step 2:
Simplify second expression, $\frac{x^2+4x-21}{9x^4}$

= $\frac{(x + 7)(x - 3)}{9x^4}$

Step 3:
Multiply both the expressions

$\frac{6x^5}{(x + 9)(x + 7)}$ * $\frac{(x + 7)(x - 3)}{9x^4}$ =  $\frac{2x(x - 3)}{3(x + 9)}$

$\frac{2x(x - 3)}{3(x + 9)}$

Question 2: Multiply and simplify $\frac{12x^6}{x^2+15x+36}$ and $\frac{x^2+10x-24}{3x^5}$
Solution:

Given expressions are $\frac{12x^6}{x^2+15x+36}$ and $\frac{x^2+10x-24}{3x^5}$

Step 1:
Simplify first expression, $\frac{12x^6}{x^2+15x+36}$

= $\frac{12x^6}{(x+12)(x + 3)}$

Step 2: Simplify second expression, $\frac{x^2+10x-24}{3x^5}$

= $\frac{(x + 12)(x - 2)}{3x^5}$

Step 3:
Multiply both the expressions

$\frac{12x^6}{(x+12)(x + 3)}$ * $\frac{(x + 12)(x - 2)}{3x^5}$

= $\frac{4x(x - 2)}{x + 3}$

$\frac{4x(x - 2)}{x + 3}$.

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