(a) Express \(\frac{2\sqrt{2}}{\sqrt{48} - \sqrt{8} - \sqrt{27}}\) in the form \(p + q\sqrt{r}\), where p, q and r are rational numbers.
(b) If \(V = A\log_{10} (M + N)\), express N in terms of M, V and A.
(b) If \(V = A\log_{10} (M + N)\), express N in terms of M, V and A.
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Correct Answer: Option n
Explanation:
(a) \(\frac{2\sqrt{2}}{\sqrt{48} - \sqrt{8} - \sqrt{27}}\)
= \(\frac{2\sqrt{2}}{\sqrt{16 \times 3} - \sqrt{4 \times 2} - \sqrt{9 \times 3}}\)
= \(\frac{2\sqrt{2}}{4\sqrt{3} - 2\sqrt{2} - 3\sqrt{3}}\)
= \(\frac{2\sqrt{2}}{\sqrt{3} - 2\sqrt{2}}\)
= \((\frac{2\sqrt{2}}{\sqrt{3} - 2\sqrt{2}})(\frac{\sqrt{3} + 2\sqrt{2}}{\sqrt{3} + 2\sqrt{2}})\)
= \(\frac{2\sqrt{6} + 4(2)}{3 + 2\sqrt{6} - 2\sqrt{6} - 4(2)}\)
= \(\frac{2\sqrt{6} + 8}{3 - 8}\)
= \(\frac{8 + 2\sqrt{6}}{-5}\)
= \(-\frac{8}{5} - \frac{2\sqrt{6}}{5}\)
= \(p = -\frac{8}{5}; q = -\frac{2}{5} ; r = 6\)
(b) \(V = A\log_{10} (M + N)\)
\(\log_{10} (M + N) = \frac{V}{A}\)
\(10^{\frac{V}{A}} = M + N \)
\(N = 10^{\frac{V}{A}} - M\)
(a) \(\frac{2\sqrt{2}}{\sqrt{48} - \sqrt{8} - \sqrt{27}}\)
= \(\frac{2\sqrt{2}}{\sqrt{16 \times 3} - \sqrt{4 \times 2} - \sqrt{9 \times 3}}\)
= \(\frac{2\sqrt{2}}{4\sqrt{3} - 2\sqrt{2} - 3\sqrt{3}}\)
= \(\frac{2\sqrt{2}}{\sqrt{3} - 2\sqrt{2}}\)
= \((\frac{2\sqrt{2}}{\sqrt{3} - 2\sqrt{2}})(\frac{\sqrt{3} + 2\sqrt{2}}{\sqrt{3} + 2\sqrt{2}})\)
= \(\frac{2\sqrt{6} + 4(2)}{3 + 2\sqrt{6} - 2\sqrt{6} - 4(2)}\)
= \(\frac{2\sqrt{6} + 8}{3 - 8}\)
= \(\frac{8 + 2\sqrt{6}}{-5}\)
= \(-\frac{8}{5} - \frac{2\sqrt{6}}{5}\)
= \(p = -\frac{8}{5}; q = -\frac{2}{5} ; r = 6\)
(b) \(V = A\log_{10} (M + N)\)
\(\log_{10} (M + N) = \frac{V}{A}\)
\(10^{\frac{V}{A}} = M + N \)
\(N = 10^{\frac{V}{A}} - M\)