Answer: (a) \(x=3\). (b) \((a,b)=(2,7)\) or \((0,3)\).

Rewrite the exponential terms in a common form, or introduce a substitution, so that the equation becomes algebraic.

Let \(u=8^{1/x}\). Then \(8^{-1/x}=\frac1u\), so the equation becomes

\(u-\frac{2}{u}=1.\)

Multiplying by \(u\),

\(u^2-u-2=0.\)

So \((u-2)(u+1)=0\). Since \(u=8^{1/x}\gt0\), \(u=2\).

Thus \(8^{1/x}=2=8^{1/3}\), giving \(x=3\).

For part (b), expand the left-hand side:

\((a-\sqrt3)^2=a^2-2a\sqrt3+3.\)

Compare this with \(b+(3-b)\sqrt3\). Equating rational and irrational parts gives

\(a^2+3=b\), and \(-2a=3-b\).

Substitute \(b=a^2+3\) into the second equation:

\(-2a=3-(a^2+3)=-a^2.\)

So \(a^2-2a=0\), and \(a(a-2)=0\). Hence \(a=0\) or \(a=2\).

If \(a=0\), then \(b=3\). If \(a=2\), then \(b=7\).

This gives the required answer: (a) \(x=3\). (b) \((a,b)=(2,7)\) or \((0,3)\).