Unit-9 | Simplification of Rational Expressions | Class-10


Overview: Expressions that do not contain surds and can be expressed in the form of p/q where q is not equal to 0 are called rational expressions. Simplification of Rational Expressions is only possible if the denominators of the given expressions are identical or same or equal.



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Rational Expressions 

Simplification of Rational Expressions


Rational Expressions:

An expression in the form of p/q where q is not equal to 0 is called a rational expression. But, the surds can not be a rational expression.

Some examples of rational expressions are: 
1. $\dfrac{a}{b}$
2. $\dfrac{a²}{a-2}$
3. $\dfrac{a+3}{a-b}$

And, expressions consisting surds can not be rational expressions such as:
1. $\sqrt{\dfrac{a}{b}}$
2. $\sqrt{\dfrac{a-b}{a+b}}$

Simplification of Rational Expressions:

We know, rational expressions always have a denominator that is not equal to zero.

When we want to add or subtract two different rational expressions then, we need to remember the following points:

  • The denominator of both the expressions should be same.
  • If the denominators of both the expressions are not same, we need to take the LCM to make them same. 
  • After this, you can add or subtract the terms in the numerator. 
  • You will combine the expressions and write only single denominator. Meaning, out of the two same denominators of the two expressions, you will only write one.
  • Then, write the expressions in the numerator as well as in the denominator in factor form. Now, divide the equal factors, if any.
  • Whatever remains is your answer. 

Click on Simplify to see some solved examples of Simplification of Rational Expressions: [Note: These links will take you to SciPiPupil, our site for solutions only.]

Simplify: $\frac{x+3}{x²+3x+9}+ \frac{x-3}{x²-3x+9}-\frac{54}{x⁴+9x²+81}$

Simplify: $\frac{x+y}{x-y}+ \frac{x-y}{x+y}-\frac{2(x²-y²)}{x²+y²}$

Simplify: $\frac{3x-1}{9x²-3x+1}- \frac{3x+1}{9x²+3x+1}+\frac{54x³}{81x⁴+9x²+1}$

Simplify: $\frac{a+2}{1+a+a²}- \frac{a-2}{1-a+a²}-\frac{2a²}{1+a²+a⁴}$

Simplify: $\frac{a+b}{a-b}- \frac{a-b}{a+b}-\frac{2ab}{a²-b²}$

Simplify: $\frac{2}{a+b}- \frac{2}{a-b}+\frac{4a}{a²-b²}$

Simplify: $\frac{2a-6}{a²-9a+20} - \frac{a-1}{a²-7a+12}-\frac{a-2}{a²-8a+15}$

Simplify: $\frac{x-1}{(2x-1)(x+2)} + \frac{3}{(x+2)(x-1)}-\frac{1}{(1-x)(1-2x)}$

Simplify: $\frac{1}{(x-3)(x+2)} + \frac{3}{(x+2)(4-x)}+\frac{2}{(x-3)(x-4)}$

Simplify: $\frac{1}{(a-b)(b-c)} + \frac{1}{(c-b)(a-c)}$

Simplify: $\frac{x-2}{x^2-1} - \frac{x+1}{x^2-2x+1}$

Simplify: $\frac{x-4}{x+6} - \frac{x-6}{x+4}$

Simplify: $\frac{p+2}{p-2} - \frac{p-2}{p+2}$

Simplify: $\frac{a+1}{a-1}+\frac{a^2-1}{a+1}$



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