Without using Mathematical tables or a calculator, simplify : (a) \(\sqrt{50} - 3\sqrt{2}(2\sqrt{2} - 5) - 5\sqrt{32}\) (b) \(\frac{1}{2} \log_{10} \frac{25...
Question 1 Report
Without using Mathematical tables or a calculator, simplify :
(a) \(\sqrt{50} - 3\sqrt{2}(2\sqrt{2} - 5) - 5\sqrt{32}\)
(b) \(\frac{1}{2} \log_{10} \frac{25}{4} - 2 \log_{10} \frac{4}{5} + \log_{10} \frac{320}{125}\).
Answer Details
(a) Simplify \(\sqrt{50}-3\sqrt{2}(2\sqrt{2}-5)-5\sqrt{32}\).
\(\sqrt{50}=5\sqrt{2}\) and \(\sqrt{32}=4\sqrt{2}\), so \(5\sqrt{32}=20\sqrt{2}\).
Expand the bracket: \(3\sqrt{2}(2\sqrt{2}-5)=6(\sqrt{2}\cdot\sqrt{2})-15\sqrt{2}=6(2)-15\sqrt{2}=12-15\sqrt{2}\).
Combine:
\[5\sqrt{2}-(12-15\sqrt{2})-20\sqrt{2}=5\sqrt{2}-12+15\sqrt{2}-20\sqrt{2}=(5+15-20)\sqrt{2}-12=-12\]\(=-12\)
(b) \(\tfrac{1}{2}\log_{10}\tfrac{25}{4}-2\log_{10}\tfrac{4}{5}+\log_{10}\tfrac{320}{125}\).
Bring the coefficients inside as powers: \(\tfrac{1}{2}\log\tfrac{25}{4}=\log\left(\tfrac{25}{4}\right)^{1/2}=\log\tfrac{5}{2}\); \(\;2\log\tfrac{4}{5}=\log\tfrac{16}{25}\).
\[\log\frac{5}{2}-\log\frac{16}{25}+\log\frac{320}{125}=\log\!\left(\frac{5}{2}\times\frac{25}{16}\times\frac{320}{125}\right)\]\(\dfrac{5}{2}\times\dfrac{25}{16}=\dfrac{125}{32}\); then \(\dfrac{125}{32}\times\dfrac{320}{125}=\dfrac{320}{32}=10\).
\[=\log_{10}10=1\]\(=1\)
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