Question 1086719
If x^4+y^4=2 and x^8+y^8=3, what is the value of x^(16)+y^(16)
<pre>{{{x^4 + y^4 = 2}}} 				  {{{x^8 + y^8 = 3}}}
{{{(x^4 + y^4)^2 = 2^2}}} ------ Squaring both sides
{{{x^8 + 2x^4y^4 + y^8 = 4}}} 
{{{x^8 + y^8 + 2x^4y^4 = 4}}} ------ Rearranging terms
{{{3 + 2x^4y^4 = 4}}} ------ Substituting 3 for {{{x^8 + y^8}}} 
{{{2x^4y^4 = 1}}} 
{{{x^4y^4 = 1/2}}} ------ eq (i)


{{{x^8 + y^8 = 3 }}}
{{{(x^8 + y^8)^2 = 3^2}}} ------ Squaring both sides
{{{x^16 + 2x^8y^8 + y^16 = 9}}} 
{{{x^16 + y^16 = 9 - 2x^8y^8}}} 
{{{x^16 + y^16 = 9 - 2(x^4y^4)^2}}}
{{{x^16 + y^16 = 9 - 2(1/2)^2}}} ------- Substituting ½ for {{{x^4y^4}}} 
{{{x^16 + y^16 = 9 - 2(1/4)}}}
{{{highlight_green(matrix(1,7, x^16 + y^16, "=", 9 - 1/2, "=", 8&1/2, or, 8.5))}}}</pre>