# Simplifying rational expressions and restrictions

### Simplifying rational expressions and restrictions

A rational expression is a fraction that its numerator and/or denominator are polynomials. In this lesson, we will first learn how to find the non-permissible values of the variable in a rational expression. Then, we will how to simplify rational expressions.

#### Lessons

$\cdot$ multiplication rule: $x^a \cdot x^b=x^{a+b}$

$\cdot$ division rule: $\frac{x^a}{x^b}=x^{a-b}$
• Introduction
Why is it important to determine the non-permissible values prior to simplifying a rational expression?

• 1.
For each rational expression:
i) determine the non-permissible values of the variable, then
ii) simplify the rational expression
a)
$\frac{{6{x^3}}}{{4x}}$

b)
$\frac{{5 - {x}}}{{{x^2} - 8x + 15}}$

c)
$\frac{{{x^2} + 13x + 40}}{{{x^2} - 25}}$

• 2.
For each rational expression:
i) determine the non-permissible values of the variable, then
ii) simplify the rational expression
a)
$\frac{{9{t^3} - 16t}}{{3{t^2} + 4t}}$

b)
$\frac{{{x^2} + 2x - 3}}{{{x^4} - 10{x^2} + 9}}$

• 3.
For each rational expression:
i) determine the non-permissible values of the variable, then
ii) simplify the rational expression
a)
$\frac{{x - 3}}{{3 - x}}$

b)
$\frac{{5{y^3} - 10{y^2}}}{{30 - 15y}}$

c)
$\frac{{1 - 9{x^2}}}{{6{x^2} - 7x - 3}}$

• 4.
The area of a rectangular window can be expressed as $4{x^2} + 13x + 3$, while its length can be expressed as $4x + 1$.
a)
Find the width of the window.

b)
If the perimeter of the window is 68 $m$, what is the value of $x$?

c)
If a cleaning company charges \$3/$m^2$ for cleaning the window, how much does it cost to clean the window?

• 5.
For each rational expression:
i) determine the non-permissible values for $y$ in terms of $x$ , then
ii) simplify, where possible.
a)
$\frac{{2x + y}}{{2x - y}}$

b)
$\frac{{x - 3y}}{{{x^2} - 9{y^2}}}$