Operate on rational expressions
When we multiply and divide rational expressions it is common that one tries to cancel terms instead of factors, this is not allowed and we must follow these rules:
$$\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}$$
$$\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}$$
$$b\neq 0,\; d\neq 0,\; c\neq 0$$
Adding and subtracting rational expressions are a little bit trickier; we must first find equivalent fractions that have a common denominator.
$$\frac{e}{f}+\frac{g}{h}=\frac{e\cdot h}{f\cdot h}+\frac{g\cdot f}{h\cdot f}=\frac{eh+gf}{hf}$$
Example
Simplify the following expression
$$\frac{x+\frac{x}{2}}{\frac{x}{2}+\frac{3x}{2}}$$
First we simplify the numerator and denominator
$$\frac{x+\frac{x}{2}}{\frac{x}{2}+\frac{3x}{2}}=$$
$$=\frac{\frac{3x}{2}}{\frac{4x}{2}}=$$
$$=\frac{\frac{3x}{2}}{\frac{2x}{1}}=$$
$$=\frac{3x}{2}\cdot {\frac{1}{2x}}=$$
$$=\frac{3x\cdot 1}{2\cdot 2x}=$$
$$=\frac{3x}{4x}$$
The last step we do is that we cancel x in both our numerator and denominator since they are factors and not terms.
$$\frac{3x}{4x}=\frac{3}{4}\; ,x\neq 0$$
Video lesson
Simplify the given expression
$$\frac{\frac{x}{6}-\frac{6}{x}}{\frac{(x+6)(x-6)}{6x}}$$