(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.
Show Answer Show Explanation 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) \(V = A\log_{10} (M + N)\) \(\log_{10} (M + N) = \frac{V}{A}\) \(10^{\frac{V}{A}} = M + N \) \(N = 10^{\frac{V}{A}} - M\)