# How do you simplify (3 + 1/(x + (1/(x + 2/x))))/(3/(x + 2))?

Aug 14, 2016

((3x^3+9x+x^2+2)(x+2))/(3x(x^2+3)

Does this simplify further?

#### Explanation:

There is a neat method which might be of use in this fraction.

$\frac{1}{\frac{x}{y}} = \frac{y}{x} \text{ " and " } \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a d}{b c}$

Let's do this bit-by-bit....

$\frac{3 + \frac{1}{x + \left(\textcolor{red}{\frac{1}{x + \frac{2}{x}}}\right)}}{\frac{3}{x + 2}}$

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Simplify the red part:
$\textcolor{red}{\frac{x}{1} + \frac{2}{x}} = \textcolor{red}{\frac{{x}^{2} + 2}{x}} \text{ } \Rightarrow \textcolor{red}{\frac{1}{\frac{{x}^{2} + 2}{x}} = \frac{x}{{x}^{2} + 2}}$

Now we have:
$\frac{3 + \frac{1}{x + \left(\textcolor{red}{\frac{x}{{x}^{2} + 2}}\right)}}{\frac{3}{x + 2}}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\frac{3 + \frac{1}{\textcolor{b l u e}{x + \left(\frac{x}{{x}^{2} + 2}\right)}}}{\frac{3}{x + 2}}$

Simplify the blue part:

$\textcolor{b l u e}{\frac{x}{1} + \frac{x}{{x}^{2} + 2} = \frac{{x}^{3} + 2 x + x}{{x}^{2} + 2} = \frac{{x}^{3} + 3 x}{{x}^{2} + 2}}$

color(blue)(1/(x + (x/(x^2+2)))= (x^2+2)/(x^3+3x)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now we have:

$\frac{3 + \textcolor{b l u e}{\frac{{x}^{2} + 2}{{x}^{3} + 3 x}}}{\frac{3}{x + 2}}$

$\frac{\textcolor{g r e e n}{\left(3 + \frac{{x}^{2} + 2}{{x}^{3} + 3 x}\right)}}{\frac{3}{x + 2}}$

Simplify the green part:
$\textcolor{g r e e n}{\left(\frac{3}{1} + \frac{{x}^{2} + 2}{{x}^{3} + 3 x}\right)}$

=color(green)((3x^3+9x+x^2+2)/(x^3+3x) = (3x^3+9x+x^2+2)/(x(x^2+3))
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Now we have
$\frac{\frac{3 {x}^{3} + 9 x + {x}^{2} + 2}{x \left({x}^{2} + 3\right)}}{\frac{3}{x + 2}}$

=((3x^3+9x+x^2+2)(x+2))/(3x(x^2+3)

Would this simplify further?