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