If 314\(_{10}\) - 256\(_7\) = 340\(_x\), find x.

Simplify \(\frac{3(2^{n+1}) - 4(2^{n-1})}{2^{n+1} - 2^n}\)

If \(P344_{6} - 23P2_{6} = 2PP2_{6}\), find the value of the digit P.

A binary operation * is defined by a * b = a\(^b\). If a * 2 = 2 - a, find the possible values of a.