Question 1046030
<pre>
Please help me solving 
If {{{a=(4(6)^(1/2))/(2^(1/2)+3^(1/2))}}} then the value of
{{{(a+2(2)^(1/2))/(a-2(2)^(1/2))+(a+2(3)^(1/2))/(a-2 (3)^(1/2))}}} is given by?

{{{a = matrix(2,1, " ", 4(6)^(1/2))/matrix(2,1, " ", 2^(1/2) + 3^(1/2))}}} 
{{{a = (4sqrt(6))/(sqrt(2) + sqrt(3))}}} --- Applying {{{matrix(2,1, " ", x^(y/z) = root(z, x^y))}}}
{{{a = (4sqrt(6)(sqrt(2) - sqrt(3)))/((sqrt(2) + sqrt(3))(sqrt(2) - sqrt(3)))}}} -- Rationalizing denominator by multiplying numerator and denominator by
                         {{{sqrt(2) - sqrt(3)}}}, the CONJUGATE of the denominator, {{{sqrt(2) + sqrt(3)}}}
{{{a = (4sqrt(12) - 4sqrt(18))/((sqrt(2))^2 - sqrt(3)sqrt(2) + sqrt(3)sqrt(2) - (sqrt(3))^2)}}} 
{{{a = (4sqrt(12) - 4sqrt(18))/(2 - 3)}}} ===> {{{a = (4sqrt(12) - 4sqrt(18))/(- 1)}}}
                       {{{a = (4sqrt(4)sqrt(3) - 4sqrt(9)sqrt(2))/(- 1)}}} ===> {{{a = (4(2)sqrt(3) - 4(3)sqrt(2))/(- 1)}}}
                                               {{{a = (8sqrt(3) - 12sqrt(2))/(- 1)}}} ===> {{{a = - 8sqrt(3) + 12sqrt(2)}}} --- Multiplying numerator and 
                                                                                       denominator by - 1

{{{highlight_green(matrix(2,1, " ", a + 2(2)^(1/2))/matrix(2,1, " ", a - 2(2)^(1/2)) + matrix(2,1, " ", (a + 2(3)^(1/2))/matrix(2,1, " ", a - 2(3)^(1/2))))}}}
{{{(a + 2sqrt(2))/(a - 2sqrt(2)) + (a + 2sqrt(3))/(a - 2sqrt(3))}}} --- Applying {{{matrix(2,1, " ", x^(y/z) = root(z, x^y))}}}

{{{(- 8sqrt(3) + 12sqrt(2) + 2sqrt(2))/(- 8sqrt(3) + 12sqrt(2) - 2sqrt(2)) + (- 8sqrt(3) + 12sqrt(2) + 2sqrt(3))/(- 8sqrt(3) + 12sqrt(2) - 2sqrt(3))}}} ----- Substituting {{{- 8sqrt(3) + 12sqrt(2)}}} for a

{{{(- 8sqrt(3) + 14sqrt(2))/(- 8sqrt(3) + 10sqrt(2)) + (- 6sqrt(3) + 12sqrt(2))/(- 10sqrt(3) + 12sqrt(2))}}} 

{{{(cross(- 2)(4sqrt(3) - 7sqrt(2)))/(cross(- 2)(4sqrt(3) - 5sqrt(2))) + (cross(- 2)(3sqrt(3) - 6sqrt(2)))/(cross(- 2)(5sqrt(3) - 6sqrt(2)))}}} ====> {{{(4sqrt(3) - 7sqrt(2))/(4sqrt(3) - 5sqrt(2)) + (3sqrt(3) - 6sqrt(2))/(5sqrt(3) - 6sqrt(2))}}} =====> {{{((4sqrt(3) - 7sqrt(2)) (5sqrt(3) - 6sqrt(2)) + (3sqrt(3) - 6sqrt(2))(4sqrt(3) - 5sqrt(2)))/((4sqrt(3) - 5sqrt(2))(5sqrt(3) - 6sqrt(2)))}}}

{{{((20sqrt(3)sqrt(3) - 24sqrt(3)sqrt(2) - 35sqrt(3)sqrt(2) + 42sqrt(2)sqrt(2)) + (12sqrt(3)sqrt(3) - 15sqrt(3)sqrt(2) - 24sqrt(3)sqrt(2) + 30sqrt(2)sqrt(2)))/((4sqrt(3) - 5sqrt(2))(5sqrt(3) - 6sqrt(2)))}}}

{{{((20(3) - 59sqrt(3)sqrt(2) + 42(2)) + (12(3) - 39sqrt(3)sqrt(2) + 30(2)))/((4sqrt(3) - 5sqrt(2))(5sqrt(3) - 6sqrt(2)))}}}

{{{((60 - 59sqrt(3)sqrt(2) + 84) + (36 - 39sqrt(3)sqrt(2) + 60))/((4sqrt(3) - 5sqrt(2))(5sqrt(3) - 6sqrt(2)))}}}

{{{(144 - 59sqrt(3)sqrt(2) + 96 - 39sqrt(3)sqrt(2))/((4sqrt(3) - 5sqrt(2))(5sqrt(3) - 6sqrt(2)))}}}

{{{(240 - 98sqrt(3)sqrt(2))/((4sqrt(3) - 5sqrt(2))(5sqrt(3) - 6sqrt(2)))}}}

{{{(240 - 98sqrt(3)sqrt(2))/((20sqrt(3)sqrt(3) - 24sqrt(3)sqrt(2) - 25sqrt(3)sqrt(2) + 30sqrt(2)sqrt(2)))}}}

{{{(240 - 98sqrt(3)sqrt(2))/(20(3) - 49sqrt(3)sqrt(2) + 30(2))}}}

{{{(240 - 98sqrt(3)sqrt(2))/(60 - 49sqrt(3)sqrt(2) + 60)}}}  

{{{(240 - 98sqrt(3)sqrt(2))/(120 - 49sqrt(3)sqrt(2))}}} ===> {{{(2(120 - 49sqrt(3)sqrt(2)))/(120 - 49sqrt(3)sqrt(2))}}}
                   {{{(2cross((120 - 49sqrt(3)sqrt(2))))/cross((120 - 49sqrt(3)sqrt(2))))}}} = <font color = red><font size  = 4><b>2</font></font></b> <=== {{{highlight_green(matrix(2,1, " ", a + 2(2)^(1/2))/matrix(2,1, " ", a - 2(2)^(1/2)) + matrix(2,1, " ", (a + 2(3)^(1/2))/matrix(2,1, " ", a - 2(3)^(1/2))))}}}, IF {{{a = matrix(2,1, " ", 4(6)^(1/2))/matrix(2,1, " ", 2^(1/2) + 3^(1/2))}}}</pre>