Question 19451
YOUR SINCERE DESIRE TO TRY BY YOUR SELF FIRST AND ASK FOR CLARRIFICATIONS IS REALLY IMPRESSIVE.YOU WILL CERTAINLY SUCCEED WITH THIS APPROACH...............MY COMMENTS ARE GIVEN IN BOLD 

x^2+x-x^2
_________________YOU BETER TYPE THIS USING BRACKETS AS FOLLOWS..IN VIEW OF sqrt x^2+x+x    ....... WHAT YOU SAID BELOW ..sqrt(x^2+x)+x


However, only the first two terms in the denominator are under the radical. 
According to this book the above equation becomes: 
x                      GOOD
___________________
x(sqrt 1+1/x+1)     TYPE AS x(sqrt(1+1/x)+1)
Again, however, the third term in the denominator is not under the radical.

Concerning the numerator, I understand why the previous equation's numerator is x. VERY GOOD

But I don't really know why and how the denominator in the previous equation is derived. The author says "Factor the x out of the denominator" but I am confused about what specific steps are needed to get the result in this denominator....OK LET ME EXPLAIN....WE FIND x IN NR,WE HAVE x BY SIDE OF SQRT TERM IN DR..SO IF WE CAN TAKE OUT x FROM SQRT TERM ALSO ,WE CAN CANCEL x THROUGH OUT...BUT TO TAKE x FROM SQUARE ROOT ,WE HAVE TO TAKE x^2 AS SQRT OF x^2=x..SO WE WRITE  sqrt(x^2+x) IN DR AS sqrt(x^2(x^2/x^2 +x/x^2))
=sqrt(x^2(1+1/x))=sqrtx^2*sqrt(1+1/x)=xsqrt(1+1/x) ..HOPE IT IS CLEAR
A math friend of mine suggested that I factor x^2 out of sqrt x^2+x under the radical. 
CORRECT
I assume that I could make the sqrt x^2 under the denominator first into sqrt (x^2)(1)
CORRECT UP TO THIS 
to get x(sqrt 1... concerning this first term, but this still leaves me figuring out how the second term becomes 1/x.NOW IT IS CLEAR I SUPPOSE
SO FINALLY WE GOT NOW IN THE DR sqrt(x^2+x)+x = xsqrt(1+1/x)+x=x(sqrt(1+1/x))+1)
=x(1+sqrt(1+1/x))...hence the given expression is 

x^2+x-x^2
_________________
sqrt x^2+x+x   
=x/x(1+sqrt(1+1/x))
= 1/(1+sqrt(1+1/x))