<s>[INST] Simplifying the expression $(1+\sqrt[4]3)/(1-\sqrt[4]3)+1/(1+\sqrt[4]3)+2/(1+\sqrt{3})$ Can anyone give provide me some help to simplify this expression?
The three denominators are pretty much different, and I can't find a common denominator.
$$\frac{1+\sqrt[4]3}{1-\sqrt[4]3}+\frac1{1+\sqrt[4]3}+\frac2{1+\sqrt{3}}$$
The calculator said it's equal to $-2$, but I don't get how a complicated expression like this would be equal to $-2$.
 [/INST] Putting  $\sqrt[4]3=x$
$$\frac{1+\sqrt[4]3}{1-\sqrt[4]3}+\frac1{1+\sqrt[4]3}+\frac2{1+\sqrt{3}}$$
$$=\frac{1+x}{1-x}+\frac1{1+x}+\frac2{1+x^2}$$
$$=\frac x{1-x}+\frac1{1-x}+\frac1{1+x}+\frac2{1+x^2}$$
$$=\frac x{1-x}+\frac{1+x+1-x}{(1-x)(1+x)}+\frac2{1+x^2}$$
$$=\frac x{1-x}+2\left(\frac1{1-x^2}+\frac1{1+x^2}\right)$$
$$=\frac x{1-x}+2\left(\frac{1+x^2+1-x^2}{(1-x^2)(1+x^2)}\right)$$
$$=\frac x{1-x}+\frac4{1-x^4}$$
Now $x^4=3$ as  $\sqrt[4]3=x$
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