Apporoaches to solve the given algebraic expression If  $\displaystyle \ \  x^{4} \ +\ x^{2} \ =\ \frac{11}{5}$  then what is the value of the given expression
$$\displaystyle \left(\frac{x+1}{x-1}\right)^{\frac{1}{3}} +\ \left(\frac{x-1}{x+1}\right)^{\frac{1}{3}} \ =\ \ ?$$
My Try :
As I can find the value of $\displaystyle x$, from the given equation but it will be tedious I think !.
$$\displaystyle x^{4} \ +\ x^{2} \ =\ \frac{11}{5}$$
$$\displaystyle \Longrightarrow \ x^{2} +1/2 \ =\ \frac{7}{2\sqrt{5}}$$
$$\displaystyle \Longrightarrow \ x^2 \ =\  \ \frac{7-\sqrt{5}}{2\sqrt{5}}$$
Which is getting too much complicated to solve the expression by putting the value of $\displaystyle x$.
What could be the other way to solve the given expression?

Let
$$ a = \displaystyle \left(\frac{x+1}{x-1}\right)^{\frac{1}{3}} +\ \left(\frac{x-1}{x+1}\right)^{\frac{1}{3}}. $$
Notice that the first term is the multiplicative inverse of the second term. Thus
$$ a^3 = \displaystyle \frac{x+1}{x-1} + \frac{x-1}{x+1} + 3 \left(\displaystyle \left(\frac{x+1}{x-1}\right)^{\frac{1}{3}} +\ \left(\frac{x-1}{x+1}\right)^{\frac{1}{3}}   \right). $$
So
$$ a^3 - 3 a =  \displaystyle \frac{x+1}{x-1} + \frac{x-1}{x+1}. $$
You can calculate the right hand side and then solve for $a$.
