(a) Solve \(\frac{1}{81^{(x – 2)}} = 27^{(1 – x)}\)
(b) Simplify \(\frac{5}{\sqrt{7} – \sqrt{3}} + \frac{1}{\sqrt{7} + \sqrt{3}}\), leaving your answer in surd form.
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) \(\frac{5}{\sqrt{7} - \sqrt{3}} + \frac{1}{\sqrt{7} + \sqrt{3}}\)
= \(\frac{5(\sqrt{7} + \sqrt{3}) + (\sqrt{7} - \sqrt{3})}{(\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3})}\)
= \(\frac{5\sqrt{7} + 5\sqrt{3} + \sqrt{7} - \sqrt{3}}{7 - 3}\)
= \(\frac{6\sqrt{7} + 4\sqrt{3}}{4}\)
= \(\frac{3\sqrt{7} + 2\sqrt{3}}{2}\).