Question 175668
{{{((u + 1)/(1 + (1/u)))/(1/(u + 1))}}}


Start by inverting the denominator fraction ( {{{1/(u + 1)}}} ) and multiplying times the numerator fraction.


{{{((u + 1)/(1 + (1/u)))((u + 1)/1) = ((u + 1)(u + 1))/(1 + (1/u))}}}


Now your denominator has addition of an integer to a fraction.  The LCD is {{{u}}}, so  {{{1 + (1/u) = (u + 1)/u}}}.  Replace the denominator with this new expression:


{{{((u + 1)(u + 1))/(1 + (1/u))=((u + 1)(u + 1))/((u+1)/u)}}}


Again, invert and multiply:


{{{(((u + 1)(u + 1))/1)(u/(u+1))}}}


Remove like terms from numerator and denominator:


{{{((cross((u + 1))(u + 1))/1)(u/cross(u+1))}}}


{{{u(u + 1)}}} or {{{u^2 + u}}} if you prefer.


Check the answer:  You can't absolutely prove the answer this way, but you can get a pretty good idea of whether you did the manipulations correctly.  Pick a number.  Best is a small whole number other than zero or one.  Let's try 2.  If u is 2, then plugging 2 into the original expression should yield 2 X (2 + 1) = 6.


{{{((2 + 1)/(1 + (1/2)))/(1/(2 + 1))}}}.  You get to do the arithmetic to see if my solution is correct.