Question 982481
x= {{{sqrt(7)+sqrt(3)}}}/{{{sqrt(7)-sqrt(3)}}} and y={{{sqrt(7)-sqrt(3)}}}/{{{sqrt(7)+sqrt(3)}}}

What? Maybe this instead?
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{{{x=(sqrt(7)+sqrt(3))/(sqrt(7)-sqrt(3))}}} and {{{y=(sqrt(7)-sqrt(3))/(sqrt(7)+sqrt(3))}}}


Using them, find what happens if substitute for question (1).


{{{y/x+x/y}}}


{{{((sqrt(7)-sqrt(3))/(sqrt(7)+sqrt(3)))/((sqrt(7)+sqrt(3))/(sqrt(7)-sqrt(3)))+((sqrt(7)+sqrt(3))/(sqrt(7)-sqrt(3)))/((sqrt(7)-sqrt(3))/(sqrt(7)+sqrt(3)))}}}


The next two steps are difficult to give through the keyboard and text and I omit them here; involving "invert and multiply" for the complex fractions and of multiplying binomials;


but then you can show the step


{{{(10-2sqrt(21))/(10+2sqrt(21))+(10+2sqrt(21))/(10-2sqrt(21))}}}, which is reducible


{{{(5-sqrt(21))/(5+sqrt(21))+(5+sqrt(21))/(5-sqrt(21))}}}


Next, the process of rationalizing the denominators will also act to raise each rational expression to higher terms having the common denominator.



{{{((5-sqrt(21))/(5+sqrt(21)))(((5-sqrt(21))/(5-sqrt(21))))+((5+sqrt(21))/(5-sqrt(21)))(((5+sqrt(21))/(5+sqrt(21))))}}}, and you have the result causing Difference of Squares for the denominators.  (NOTE that this step is now  fixed...)


{{{(25-10sqrt(21)+21)/(25-21)+(25+10sqrt(21)+21)/(25-21)}}}


The rest of the arithmetic steps should be expected without overly-strategizing.


{{{(50+42)/4}}}


{{{92/4}}}


{{{highlight(highlight(23))}}}