Question 103185
1.{{{(1+ 2/x+ 4/x2+ 8/x3)/(1- 16/x4)}}}
Multiply top and bottom by {{{x^4}}}
{{{(x^4+ 2x^3+ 4x^2+ 8x)/(x^4-16)}}}
{{{x(x^3+ 2x^2+ 4x+ 8)/(x^4-16)}}}
Let's look at the denominator first.
{{{(x^4-16)=(x^2+4)(x^2-4)}}}
{{{(x^4-16)=(x^2+4)(x+2)(x-2)}}}
Now the numerator
{{{x(x^3+2x^2+4x+8))=x(x^2+4)(x+2)}}}
Let's substitute
{{{x(x^3+ 2x^2+ 4x+ 8)/(x^4-16)}}}
{{{(x*cross((x^2+4))*cross((x+2)))/(cross((x^2+4))*cross((x+2))*(x-2))}}}
{{{x/(x-2)}}}
Good idea to check the answer
Let's try x=3
Equation 1 gives an answer 
1.Numerator={{{(1+ 2/3+ 4/9+ 8/27)}}}
1.Denominator={{{(1- 16/81)}}}
Numerator = 2.470407
Denominator = 0.802469
Numerator/Denominator = 3
{{{x/(x-2)=3/(3-2)}}}
{{{x/(x-2)=3}}}
3=3 
Good answer. 
{{{(1+ 2/x+ 4/x2+ 8/x3)/(1- 16/x4)=x/(x-2)}}}