(a) Simplify, without using Mathematical tables: (log_{10} (frac{30}{16}) – 2 log_{10} (frac{5}{9}) + log_{10} (frac{400}{243}))
(b) Without using Mathematical tables, calculate (sqrt{frac{P}{Q}}) where (P = 3.6 times 10^{-3}) and (Q = 2.25 times 10^{6}), leaving your answer in standard form.
Explanation
(a) (log_{10} (frac{30}{16}) – 2 log_{10} (frac{5}{9}) + log_{10} (frac{400}{243}))
= (log_{10} (frac{30}{16}) – log_{10} (frac{5}{9})^{2} + log_{10} (frac{400}{243}))
= (log_{10} (frac{30}{16} times frac{400}{243} div frac{25}{81}))
= (log_{10} (frac{30}{16} times frac{400}{243} times frac{81}{25}))
= (log_{10} (10) = 1)
(b) (sqrt{frac{P}{Q}} = sqrt{frac{3.6 times 10^{-3}}{2.25 times 10^{6}})
= (sqrt{frac{36 times 10^{-4}}{2.25 times 10^{6}})
= (frac{6 times 10^{-2}}{1.5 times 10^{3}})
= (4 times 10^{-2 – 3} = 4 times 10^{-5})