(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_...
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(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.
Answer Details
(a) Simplify the surds: \(\sqrt{48} = 4\sqrt{3},\ \sqrt{8} = 2\sqrt{2},\ \sqrt{27} = 3\sqrt{3}\).
\[\sqrt{48} - \sqrt{8} - \sqrt{27} = 4\sqrt{3} - 2\sqrt{2} - 3\sqrt{3} = \sqrt{3} - 2\sqrt{2}\]So the expression is \(\dfrac{2\sqrt{2}}{\sqrt{3} - 2\sqrt{2}}\). Rationalise using \((\sqrt{3} + 2\sqrt{2})\):
\[\frac{2\sqrt{2}(\sqrt{3} + 2\sqrt{2})}{(\sqrt{3})^2 - (2\sqrt{2})^2} = \frac{2\sqrt{6} + 8}{3 - 8} = \frac{2\sqrt{6} + 8}{-5}\] \[= -\frac{8}{5} - \frac{2}{5}\sqrt{6}\]This is of the form \(p + q\sqrt{r}\) with \(p = -\tfrac{8}{5},\ q = -\tfrac{2}{5},\ r = 6\).
(b) From \(V = A\log_{10}(M + N)\):
\[\frac{V}{A} = \log_{10}(M + N) \ \Rightarrow\ M + N = 10^{V/A}\] \[N = 10^{V/A} - M\]Download The App On Google Playstore
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