Question 1117168
<pre>
solve Sqrt {2+sqrt {3}}

<s>solve</s> SIMPLIFY {{{sqrt(2 + sqrt(3))}}}

{{{sqrt(2 + sqrt(3))}}} = {{{sqrt(1(2 + sqrt(3)))}}} = {{{sqrt((2(2 + sqrt(3)))/2)}}} ----- Multiplying RADICAND, {{{2 + sqrt(3)}}} by 1 = {{{2/2}}}
                        = {{{sqrt((4 + 2sqrt(3))/2)}}}
                        = {{{sqrt((3 + 1 + 2sqrt(3*1))/2)}}} ----- Replacing 4 with 3 + 1, and 3 with 3*1
                        = {{{sqrt((3 + 1 + 2sqrt(3)*sqrt(1))/2)}}}
                        = {{{sqrt(((sqrt(3))^2 + (sqrt(1))^2 + 2sqrt(3)*sqrt(1))/2)}}} ---- Converting 3 to {{{(sqrt(3))^2}}}, and 1 to {{{(sqrt(1))^2}}}
The numerator, {{{(sqrt(3))^2 + (sqrt(1))^2 + 2sqrt(3)*sqrt(1)}}} is in the form: {{{(a + b)^2}}}, with a being {{{sqrt(3)}}}, and b being {{{sqrt(1)}}}

                      So, {{{sqrt(((sqrt(3))^2 + (sqrt(1))^2 + 2sqrt(3)*sqrt(1))/2)}}} now becomes:
                          {{{sqrt(((sqrt(3) + sqrt(1))^2)/2)}}} 
                          {{{sqrt((sqrt(3) + sqrt(1))^2)/sqrt(2))}}} --- Applying {{{sqrt(a/b) = sqrt(a)/sqrt(b)}}}
                          {{{(sqrt(3) + sqrt(1))/sqrt(2)}}} ----- Cancelling square and sqrt in NUMERATOR
                          {{{((sqrt(3) + sqrt(1))/sqrt(2)) * (sqrt(2)/sqrt(2))}}} ------ Rationalizing DENOMINATOR
                          {{{sqrt(2)(sqrt(3) + sqrt(1))/(sqrt(2)*sqrt(2)))}}} = {{{highlight_green(highlight((sqrt(6) + sqrt(2))/2))}}}</pre>