Complex Numbers - De Moivre's Theorem

Express \(z\) where \(z=\frac{10-2\sqrt{3}\mathbf{i}}{1-3\sqrt{3}\mathbf{i}}\) in the form \(x+y\mathbf{i}\), where \(x\) and \(y\) are real.
Without the use of a G.C., prove that \({{z}^{21}}\) is a real number.