How To Simplify Algebraic Expressions Involving Common Denominators?

 

By Robert O

Algebraic fractions are easy to simplify just like fractions involving real numbers. The simplification process is even shorter when the complex fractions have a common denominator. In such a case, we only need to remove the denominator by multiplying both the top and bottom by it. The remaining step is just simplification.

How to simplify complex algebraic fractions with a common denominator?

You can approach this problem in many ways, but canceling the denominators and simplifying the resulting single fraction is the simplest. Let’s see how that works using some examples.

Example 1:

Simplify:

    \[\frac{\frac{x^{2}+4 x+16}{x+8}}{\frac{x^{2}-16}{x+8}} \mathrm{~m}\]

Solution

A quick look tells us that the complex fraction in the example involves a common denominator. These two denominators cancel out, and we get a single fraction.

    \[\frac{\frac{x^{2}+4 x+16}{x+3}}{\frac{x^{2}-16}{x+3}}=\frac{x^{2}+4 x+16}{x^{2}-16}\]

Using the algebraic identities, we can factorize and simplify the fraction.

    \[(a+b)(a+b)=a^{2}+a b+b a+b^{2}=a^{2}+2 a b+b^{2} \ldots \ldots \ldots . i\]

    \[(a-b)(a-b)=a^{2}-a b-b a+b^{2}=a^{2}-2 a b+b^{2} \ldots \ldots \ldots i i\]

    \[(a-b)(a+b)=a^{2}+a b-b a-b^{2}=a^{2}-b^{2} \ldots \ldots \ldots \ldots \ldots i i i\]

    \[\frac{x^{2}+4 x+16}{x^{2}-16}=\frac{(x+4)(x+4)}{(x-4)(x+4)}=\frac{x+4}{x-4}\]

Example 2:

Simplify:

    \[\frac{\frac{2 x^{2}}{x+4}}{\frac{8 x}{x+4}}\]

Solution

We drop the denominators since they are the same. Our Complex fraction simplifies to:

    \[\frac{\frac{2 x^{2}}{x+4}}{\frac{8 x}{x+4}}=\frac{2 x^{2}}{8 x}=\frac{x}{4}\]

Example 3:

Simplify:

    \[\frac{1-\frac{7}{x+1}}{\frac{4}{x+1}+1}\]

Solution

We cannot readily see that this complex fraction involves a common denominator until we carry out the operations. In this case, we first solve the numerator and denominator.

    \[1-\frac{7}{x+1}=\frac{(x+1)-7}{x+1}=\frac{x-6}{x+1}, \text { denominator is } x+1\]

    \[\frac{4}{x+1}+1=\frac{4+x+1}{x+1}=\frac{5+x}{x+1}, \text { denominato is } x+1\]

It is now clear that the fraction has a common denominator, x+1. These cancel out, and the result is a single fraction.

    \[\frac{\frac{x-6}{x+1}}{\frac{5+x}{x+1}}=\frac{x-6}{5+x}\]

The fraction is already in its simplest form.

Remarks

It is easier to solve complex algebraic fractions involving common denominators than those with different denominators. The first step of solving the latter type of algebraic expressions involves making the fractions have a common denominator. That is possible through the multiplication of top and bottom by the LCM of the denominators. Regardless of how complex an algebraic expression is, simplification is still possible.

About the Author

This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.