Example 1:

$$ \text{Simplify}\space\frac{x^6-729}{x^3-27} $$

Remember the formulas below; they are very helpful in simplification:

So, the given expression can be simplified accordingly:

$$ \frac{x^6-729}{x^3-27}=\frac{(x^2-9)(x^4+9x^2+81)}{(x-3)(x^2+3x+9)}=\frac{(x+3)(x-3)(x^4+9x^2+81)}{(x-3)(x^2+3x+9)} $$

This is simplified further after the x-3 is cancelled...

$$ \frac{(x+3)(x^4+9x^2+81)}{(x^2+3x+9)}=\frac{x^5+3x^4+9x^3+27x^2+81x+243}{x^2+3x+9} $$