(a) If (9^{2x – 1} = frac{81^{x – 2}}{3^{x}}), find x.
(b) Without using Mathematical Tables, evaluate: (sqrt{frac{0.81 times 10^{-5}}{2.25 times 10^{7}}})
Explanation
(a) (9^{2x – 1} = frac{81^{x – 2}}{3^{x}})
((3^{2})^{2x – 1} = frac{(3^{4})^{x – 2}}{3^{x}})
(3^{4x – 2} = frac{3^{4x – 8}}{3^{x}})
(3^{4x – 2} = 3^{4x – 8 – x})
(3^{4x – 2} = 3^{3x – 8})
(therefore 4x – 2 = 3x – 8 implies 4x – 3x = -8 + 2)
(x = -6)
(b) (sqrt{frac{0.81 times 10^{-5}}{2.25 times 10^{7}}})
= (sqrt{frac{0.81}{2.25}} times sqrt{10^{-5 – 7}})
= (frac{0.9}{1.5} times sqrt{10^{-12}})
= (0.6 times 10^{-6})
= (6 times 10^{-7})