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\(3\sqrt{45}-12\sqrt{5}+16\sqrt{20}\) 3\(\sqrt{5\times 9}-12\sqrt{5}+16\sqrt{4\times 5}\) 3\(\times 3\sqrt{5}-12\sqrt{5}+16\times 2\sqrt{5}\) 9\(\sqrt{5}-12\sqrt{5}+32\sqrt{5}\) 29\(\sqrt{5}\)

\(8^{x+1}=\frac{1}{4}\\ 2^{3(x+1)}=2^{-2}\Rightarrow 3(x+1)=-2\Rightarrow 3x+3=-2\\ 3x=-5\Rightarrow x = -\frac{5}{3}\)

2\(^3\) x 3 x 5\(^2\) 2 x 3\(^2\) x 5 2\(^2\) x 3\(^3\) x 5 The L.C.M = 2\(^2\) x 3\(^3\) x 5\(^2\)

\(\frac{0.3 \times 10^5}{0.4 \times 10^7} = \frac{3 \times 10^{-1} \times 10^{5}}{4 \times 10^{-1} \times 10^7}\) \(\frac{3 x 10^4}{4 x 10^4}\) \(\frac{3}{4}\) x 104-6 = 0.75 x 10-2 = 7.5 x 10-3

3 \(\sqrt{45} - 12\sqrt{5} + 16\sqrt{20}\) = 3 x \(\sqrt{9 \times 5} - 12 \times \sqrt{5} + 16 \times \sqrt{4 \times 5}\) = 3 x 3 x \(\sqrt{5} - 12 \times \sqrt{5} + 16 \times 2 \times \sqrt{5}\) = 9\(\sqrt{5} - 2 \sqrt{5} + 32 \sqrt{5}\) = 9\(\sqrt{5} + 32\sqrt{5} - 12\sqrt{5}\) = 29\(\sqrt{5}\)

\(\frac{3x^3}{(3x)^3}\) = \(\frac{3 \times x^3}{3^3 \times x^3}\) = \(\frac{3 \times x^3}{3 \times 3 \times 3 \times x^3}\) = \(\frac{1}{3^2}\) = \(\frac{1}{9}\)

log\(_{10}\) a + log\(_{10}\)a\(^2\) = 0.9030 log (a x a\(^2\)) = 0.9030 log a\(^3\) = 0.9030 a\(^3\) = 10^.9030(antilogarithm table) a\(^3\) = 7.998 a = 3\(\sqrt{7.998}\) a = 1.9998 ≈ 2.0

By Pythagoras theorem h2 = (2\(\sqrt{3})^2 - (\sqrt{3})^2\) = 22(\(\sqrt{3})^2 - 3\) 4(3) - 3 = 112 - 3 h2 = 9 h = \(\sqrt{9}\) = 3cm

2\(^2\) x 3\(^3\) x 5\(^1\) x k; 2\(^2\) x 3\(^2\) x 3 x 5 x k 2\(^2\) x 3\(^2\) x 15 x k smallest value for k 2\(^2\) x 3\(^2\) x 15 = 2\(^2\) x 3\(^2\) x 15\(^2\) k = 15

(a) \(\frac{3\frac{1}{12} + \frac{7}{8}}{2\frac{1}{4} - \frac{1}{6}}\) \(3\frac{1}{12} + \frac{7}{8} = \frac{37}{12} + \frac{7}{8}\) = \(\frac{74 + 21}{24}\) = \(\frac{95}{24}\) \(2\frac{1}{4} - \frac{1}{6} = \frac{9}{4} - \frac{1}{6}\) = \(\frac{27 - 2}{12}\) = \(\frac{25}{12}\) \(\therefore \frac{3\frac{1}{12} + \frac{7}{8}}{2\frac{1}{4} - \frac{1}{6}} = \frac{95}{24} \div \frac{25}{12}\) \(\frac{95}{24} \times \frac{12}{25} = \frac{19}{10}\) = \(1.9\) (b)(i) \(p = \frac{m}{2} - \frac{n^{2}}{5m}\) \(\frac{n^{2}}{5m} = \frac{m}{2} - p\) \(n^{2} = 5m(\frac{m}{2} - p)\) \(n = \pm \sqrt{5m(\frac{m}{2} - p)}\) (ii) When p = 14 and m = -8, \(n = \sqrt{5(-8)(\frac{-8}{2} - 14)}\) \(n = \sqrt{-40(- 4 - 14)}\) \(n = \sqrt{720}\) \(n = 12\sqrt{5}\) = \(\pm 26.83\)

(a) No Log 0.004592 \(\bar{3}.6620\) \((0.004592)^{\frac{1}{3}}\) \(\frac{\bar{3}.6620}{3} = \bar{1}.2207\) Antilog of \(\bar{1}.2207 = 0.1663\) (b) \(\log_{10} y + 3 \log_{10} x = 2\) \(\log_{10} y + \log_{10} x^{3} = 2\) \(\log_{10} (yx^{3}) = 2\) \(yx^{3} = 10^{2}\) \(yx^{3} = 100\) \(y = \frac{100}{x^{3}}\) (c) \(3x - 2y = 21 .... (1)\) \(4x + 5y = 5 ..... (2)\) Multiply (1) by 4 and (2) by 3, \(12x - 8y = 84 .... (1a)\) \(12x + 15y = 15 ... (2a)\) (1a) - (2a), \(- 8y - 15y = 84 - 15\) \(- 23y = 69 \implies y = -3\) \(3x - 2y = 21 \implies 3x - 2(-3) = 21\) \(3x + 6 = 21 \implies 3x = 15\) \(x = 5\)

m : n = 13 : 11 m\(^2\) - n\(^2\) : (m + n)2 = 13\(^2\) - 11\(^2\) : (13 + 11)\(^2\) = 169 - 121 : 24\(^2\) = 48 : 576 = 1 : 12

\(3^{-4}\times 9^2 \times 81^{-1}\) =\(log_9 (3^{-4}\times 9^2 \times 81^{-1})\) = \(log_9 \left(\frac{1}{3^4}\times 9^2 \times \frac{1}{81}\right)\) = \(log_9 \left(\frac{1}{81}\times \frac{81}{1}\times \frac{1}{81}\right)\) = \(log_9 \frac{1}{81}\) = \(log_9 \frac{1}{9^2}\) = \(log_9 9^{-2}\) = -2\(log_9 9\) - 2 \(\times 1\) = -2

(K2)\(_6\) * 3\(_6\) = 3\(_5\)(K4)\(_5\) (K*6\(^1\) + 2*6\(^0\)) 3*6\(^0\) = 3*5\(^0\)(K*5\(^1\) + 4*5\(^0\)) 3(6K + 2) = 3(5K + 4) 6K + 2 = 5K + 4 6K - 5K = 4 - 2 K = 2

\(\frac{2}{6-5\sqrt{3}} = \frac{2}{6-5\sqrt{3}} \times \frac{6+5\sqrt{3}}{6+5\sqrt{3}}\\ =\frac{2(6+5\sqrt{3})}{(6-5\sqrt{3})(6+5\sqrt{3})}\\ =\frac{12+10\sqrt{3}}{36-25(3)}\\ =\frac{12+10\sqrt{3}}{36-75}\\ =\frac{12+10\sqrt{3}}{-39}\\ =-\left(\frac{12}{39}+\frac{10\sqrt{3}}{39}\right)\)