Without using Mathematical tables or a calculator, simplify : (a) \(\sqrt{50} - 3\sqrt{2}(2\sqrt{2} - 5)

Explanation

(a) \(\sqrt{50} - 3\sqrt{2}(2\sqrt{2} - 5) - 5\sqrt{32}\)

\(\sqrt{25 \times 2} - (3 \times 2)\sqrt{2 \times 2} + (3 \times 5)\sqrt{2} - 5\sqrt{16 \times 2}\)

= \(5\sqrt{2} - 6(2) + 15\sqrt{2} - 5(4\sqrt{2})\)

= \(5\sqrt{2} - 12 + 15\sqrt{2} - 20\sqrt{2}\)

= \((5 + 15 - 20)\sqrt{2} - 12\)

= -12.

(b) \(\frac{1}{2} \log_{10} (\frac{25}{4}) - 2 \log_{10} (\frac{4}{5}) + \log_{10} (\frac{320}{125})\)

= \(\log_{10} (\frac{25}{4})^{\frac{1}{2}} - \log_{10} (\frac{4}{5})^{2} + \log_{10} (\frac{320}{125})\)

= \(\log_{10} (\frac{5}{2}) - \log_{10} (\frac{16}{25}) + \log_{10} (\frac{320}{125})\)

= \(\log_{10} (\frac{\frac{5}{2} \times \frac{320}{125}}{\frac{16}{25}}\)

= \(\log_{10} (\frac{\frac{160}{25}}{\frac{16}{25}}\)

= \(\log_{10} 10 = 1\)