SOLUTION: {{{x^4+y^4=2}}} and {{{x^8+y^8=3}}} then: {{{x^16+y^16=8&1/2}}} how does it do so?

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: {{{x^4+y^4=2}}} and {{{x^8+y^8=3}}} then: {{{x^16+y^16=8&1/2}}} how does it do so?      Log On


   

Question 213425: x%5E4%2By%5E4=2
and
x%5E8%2By%5E8=3
then:
x%5E16%2By%5E16=8%261%2F2
how does it do so?
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E4%2By%5E4=2
and
x%5E8%2By%5E8=3
then:
x%5E16%2By%5E16=8%261%2F2
how does it do so?
-----------------------

 
x%5E4%2By%5E4=2
 
Square both sides:
 
%28x%5E4%2By%5E4%29%5E2=2%5E2
 
x%5E8%2B2x%5E4y%5E4%2By%5E8=4

%28x%5E8%2By%5E8%29%2B2x%5E4y%5E4=4

but since x%5E8%2By%5E8=3, we get

3%2B2x%5E4y%5E4=4  or subtracting 3 
from both sides

2x%5E4y%5E4=1

divide both sides by 2:

x%5E4y%5E4=1%2F2

------------

Now we turn to 

x%5E8%2By%5E8=3
 
Square both sides:
 
%28x%5E8%2By%5E8%29%5E2=3%5E2
 
x%5E16%2B2x%5E8y%5E8%2By%5E16=9

%28x%5E16%2By%5E16%29%2B2x%5E8y%5E8=9

Isolate x%5E16%2By%5E16

x%5E16%2By%5E16=9-2x%5E8y%5E8

Rewrite x%5E8y%5E8 as%28x%5E4y%5E4%29%5E2

x%5E16%2By%5E16=9-2%28x%5E4y%5E4%29%5E2

but since x%5E4y%5E4=1%2F2, we get

x%5E16%2By%5E16=9-2%281%2F2%29%5E2

x%5E16%2By%5E16=9-2%281%2F4%29

x%5E16%2By%5E16=9-1%2F2

x%5E16%2By%5E16=8%261%2F2