Answer: \(x=\sqrt[3]{\frac12}\); \(n=10\); the 6th term is \(8064\).

Use the binomial theorem and identify the required term carefully from the general term. Keep the powers of the variable explicit so that the required coefficient or constant term can be read off clearly.

First, \((x+x^2)^8=x^8(1+x)^8\).

The 3rd term is \({}^8C_2x^{10}=28x^{10}\), and the 6th term is \({}^8C_5x^{13}=56x^{13}\).

Since these are equal,

\(28x^{10}=56x^{13}\).

So \(x^3=\frac12\), giving \(x=\sqrt[3]{\frac12}\).

For \(\left(x+\frac{2}{x}\right)^n\), the 6th term is

\({}^nC_5x^{n-5}\left(\frac{2}{x}\right)^5={}^nC_5 2^5 x^{n-10}.\)

For this term to be constant, \(n-10=0\), so \(n=10\).

The 6th term is therefore

\({}^{10}C_5\cdot2^5=252\cdot32=8064.\)

This gives the required answer: \(x=\sqrt[3]{\frac12}\); \(n=10\); the 6th term is \(8064\).