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.

$$\frac{e}{f}+\frac{g}{h}=\frac{e\cdot h}{f\cdot h}+\frac{g\cdot f}{h\cdot f}=\frac{eh+gf}{hf}$$

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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.

$$\frac{3x}{4x}=\frac{3}{4}\; ,x\neq 0$$

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Video lesson

Simplify the given expression

$$\frac{\frac{x}{6}-\frac{6}{x}}{\frac{(x+6)(x-6)}{6x}}$$