(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{5sqrt{7} + 5sqrt{3} + sqrt{7} – sqrt{3}}{7 – 3})
= (frac{6sqrt{7} + 4sqrt{3}}{4})
= (frac{3sqrt{7} + 2sqrt{3}}{2}).