Without using Mathematical tables or a calculator, simplify :
(a) (sqrt{50} – 3sqrt{2}(2sqrt{2} – 5) – 5sqrt{32})
(b) (frac{1}{2} log_{10} frac{25}{4} – 2 log_{10} frac{4}{5} + log_{10} frac{320}{125}).
Explanation
(a) (sqrt{50} – 3sqrt{2}(2sqrt{2} – 5) – 5sqrt{32})
(sqrt{25 times 2} – (3 times 2)sqrt{2 times 2} + (3 times 5)sqrt{2} – 5sqrt{16 times 2})
= (5sqrt{2} – 6(2) + 15sqrt{2} – 5(4sqrt{2}))
= (5sqrt{2} – 12 + 15sqrt{2} – 20sqrt{2})
= ((5 + 15 – 20)sqrt{2} – 12)
= -12.
(b) (frac{1}{2} log_{10} (frac{25}{4}) – 2 log_{10} (frac{4}{5}) + log_{10} (frac{320}{125}))
= (log_{10} (frac{25}{4})^{frac{1}{2}} – log_{10} (frac{4}{5})^{2} + log_{10} (frac{320}{125}))
= (log_{10} (frac{5}{2}) – log_{10} (frac{16}{25}) + log_{10} (frac{320}{125}))
= (log_{10} (frac{frac{5}{2} times frac{320}{125}}{frac{16}{25}})
= (log_{10} (frac{frac{160}{25}}{frac{16}{25}})
= (log_{10} 10 = 1)