Explanation

(a) \(\frac{1}{81^{(x - 2)}} = 27^{(1 - x)}\)

\(81^{-(x - 2)} = 27^{(1 - x)}\)

\(3^{4[-(x - 2)]} = 3^{3(1 - x)}\)

\(3^{-4(x - 2)} = 3^{3(1 - x)}\)

\(-4(x - 2) = 3(1 - x)\)

\(-4x + 8 = 3 - 3x\)

\(8 - 3 = -3x + 4x \implies x = 5\)

(b) \(\frac{5}{\sqrt{7} - \sqrt{3}} + \frac{1}{\sqrt{7} + \sqrt{3}}\)

= \(\frac{5(\sqrt{7} + \sqrt{3}) + (\sqrt{7} - \sqrt{3})}{(\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3})}\)

= \(\frac{5\sqrt{7} + 5\sqrt{3} + \sqrt{7} - \sqrt{3}}{7 - 3}\)

= \(\frac{6\sqrt{7} + 4\sqrt{3}}{4}\)

= \(\frac{3\sqrt{7} + 2\sqrt{3}}{2}\).

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