# SIMPLIFICATION OF ALGEBRAIC FRACTION

A fraction with algebra at its denominator or numerator is called an algebraic fraction.

To simplify an algebraic fraction is to write it on the lowest form possible such that they will not be any common factor.

The next examples illustrate this

Example 1
Simplify $\frac{abc}{adc}$

Solution
$\frac{\require{cancel}\bcancel{a\times c}\times b}{\require{cancel}\bcancel{a\times c}\times d}$
$\frac{b}{d}$

Example 2
Write $\frac{c^2-d^2}{c+d}$

Solution
$c^2-d^2$ is a difference of two squares.
$\frac{\require{cancel}\bcancel{(c+d)}(c-d)}{\require{cancel}\bcancel{(c+d)}}$
$c-d$

Example 3
Simplify $\frac{84x^2y^2z}{105xz^3}$

Solution:
$\frac{84}{105}\times\frac{x^2}{x}\times\frac{y^2}{1}\times\frac{z}{z^3}$
$\frac{4}{5}\times x \times y \times\frac{1}{z^2}$
$\frac{4xy^2}{5z^2}$

Example 4
Simplify $\frac{50x^2yz^3}{150x^3y^2z^4}$

Solution:
The above can be represented this way
$\frac{50}{150}\times\frac{x^2}{x^3}\times\frac{y}{y^2}\times\frac{z^3}{z^4}$
$\frac{1}{3}\times\frac{1}{x}\times\frac{1}{y}\times\frac{1}{z}$
$\frac{1}{3xyz}$

Example 5
Express $\frac{(x^2-y^2)(a^2-b^2)}{(a-b)(x+y)}$ in the simple form possible.

Solution:
This can be rearranged in this way
$\frac{x^2-y^2}{x+y}\times\frac{a^2-b^2}{a+b}$

$x^2-y^2$ and $a^2-b^2$ are differences of two squares
$\frac{\require{cancel}\bcancel{(x+y)}(x-y)}{\require{cancel}\bcancel{(x+y)}}\times\frac{\require{cancel}\bcancel{(a-b)}(a+b)}{\require{cancel}\bcancel{(a-b)}}$
$(x-y)(a+b)$.

We will continue our discussions of algebraic fraction when we add algebraic fractions

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