Question

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

Answer 1

`((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.

`1/(x/y) = y/x" " and " "(a/b)/(c/d) = (ad)/(bc)`

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

`(3 + 1/(x + (color(red)(1/(x + 2/x)))))/(3/(x + 2))`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Simplify the red part:
`color(red)(x/1 + 2/x) = color(red)((x^2+2)/x)" " rArr color(red) (1/((x^2+2)/x) = x/(x^2+2))`

Now we have:
`(3 + 1/(x + (color(red)(x/(x^2+2)))))/(3/(x + 2))`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`(3 + 1/color(blue)(x + (x/(x^2+2))))/(3/(x + 2))`

Simplify the blue part:

`color(blue)[x/1 + x/(x^2+2) = (x^3 + 2x+x)/(x^2+2) = (x^3 + 3x)/(x^2+2)]`

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

Now we have:

`[3 + color(blue)((x^2+2)/(x^3+3x))]/(3/(x + 2))`

`color(green)((3 + (x^2+2)/(x^3+3x))]/(3/(x + 2))`

Simplify the green part:
`color(green)((3 /1+ (x^2+2)/(x^3+3x))]`

=`color(green)((3x^3+9x+x^2+2)/(x^3+3x) = (3x^3+9x+x^2+2)/(x(x^2+3))`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Now we have
`((3x^3+9x+x^2+2)/(x(x^2+3)))/(3/(x + 2))`

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

Would this simplify further?