\( =\frac{8.41 \times 10^{4}}{4.3 \times 10^{5}} \\ =\frac{841 \times 10^{2}}{43 \times 10^{4}} \\ =19.558 \times 10^{-2} \\ =1.9558 \times 10^{-1} \)

\( =1.02 \times 10^{10} \times 1.20 \times 10^{-3} \\ =1.224 \times 10^{7} \)

\( =1.598 \times 10^{17}-4.58 \times 10^{15} \\ =\left(1.598-4.58 \times 10^{-2}\right) 10^{17} \\ =(1.598-0.0458) \times 10^{17} \\ =1.5522 \times 10^{17} \)

\( \left(2.75 \times 10^{7}\right)+ \left(1.23 \times 10^{8}\right) \\ =\left(2.75 \times 10^{-1}+1.23\right) \times 10^{8} \\ =(0.275+1.23) \times 10^{8} \\ =1.505 \times 10^{8} \)

(i) 3.459 \(\times\) 10⁶ = 3459000
(ii) 5.678 \(\times\)10⁴ = 56780
(iii) 1.00005 \(\times\) 10^-5 = 0.0000100005
(iv) 2.530009 \(\times\) 10^-7 = 0.0000002530009

(i) 569430000000 = 5.6943 \(\times\)10¹¹
(ii) 2000.57 = 2.00057 \(\times\) 10³
(iii) 0.0000006000 = 6.0 \(\times\) 10^-7
(iv) 0.0009000002 = 9.000002 \(\times\) 10^-4

\(\frac { 8-5\sqrt { 2 } }{ 3-2\sqrt { 2 } } =\frac{(8-5\sqrt{2})\times(3+2\sqrt{2})}{(3-2\sqrt{2})\times(+2\sqrt{2})}=\frac{24-15\sqrt{2}+16\sqrt{2}-10\times2}{3^2-(2\sqrt{2})^2}\)
=\(\frac{24+\sqrt{2}-20}{9-4\times2}=\frac{4+\sqrt{2}}{1}\) = 4 + 1.414 = 5.414

\( x^{2} =(\sqrt{5}+2)^{2}=5+4+4 \sqrt{5}=9+4 \sqrt{5} \)
\(\frac{1}{x^{2}} =\frac{1}{9+4 \sqrt{5}} \)
\( =\frac{1}{9+4 \sqrt{5}} \times \frac{9-4 \sqrt{5}}{9-4 \sqrt{5}}=\frac{9-4 \sqrt{5}}{81-80} \)
\( ^{2}+\frac{1}{x^{2}} =9+4 \sqrt{5}+9-4 \sqrt{5} \)
\( =9+9+4 \sqrt{5}-4 \sqrt{5} \)
= 18

(i) \(\frac{1}{\sqrt{50}}=\frac{1}{\sqrt{50}}\times\frac{\sqrt{50}}{\sqrt{50}}=\frac{\sqrt{50}}{\sqrt{50}}=\frac{\sqrt{5\times5\times2}}{5\times5\times2}=\frac{ ̶5̶\sqrt{2}}{ ̶5̶\times5\times2}=\frac{\sqrt{2}}{10}\)
(ii) \(\frac{5}{3\sqrt{5}}=\frac{5}{3\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}}=\frac{ ̶5̶\sqrt{5}}{3\times ̶5̶}=\frac{\sqrt{5}}{3}\)
(iii) \(\frac{\sqrt{75}}{\sqrt{18}}=\frac{\sqrt{3\times5\times5}}{\sqrt{3\times2\times3}}=\frac{5\sqrt{3}\times\sqrt{2}}{3\sqrt{2}\times\sqrt{2}}=\frac{5\sqrt{6}}{3\times2}=\frac{5\sqrt{6}}{6}\)
(iv) \(\frac{3\sqrt{5}}{\sqrt{6}}=\frac{3\sqrt{5}\times\sqrt{6}}{\sqrt{6}\times\sqrt{6}}=\frac{ ̶3̶\sqrt{30}}{ ̶6̶}=\frac{\sqrt{30}}{2}\)

\(\frac { \sqrt { 7 } -2 }{ \sqrt { 7 } +2 } =a\sqrt{7}+b\)
L.H.S = \(\frac{\sqrt{7}-2\times\sqrt{7}-2}{\sqrt{7}+2\times\sqrt{7}-2}=\frac{(\sqrt{7}-2)^2}{\sqrt{7}^2-2^2}=\frac{\sqrt{7}^2-2\sqrt{7}\times2+2^2}{7-4}\)
=\(\frac{7-4\sqrt{7}+4}{3}=\frac{11-4\sqrt{7}}{3}=\frac{11}{3}-\frac{4\sqrt{7}}{3}\)
\(\frac{-4\sqrt{7}}{3}+\frac{11}{3}=a\sqrt{7}+b\)
ஃa √7=\(\frac{-4\sqrt{7}}{3}\)
a = \(\frac{-4}{3}\)
b = \(\frac{11}{3}\)

\(\frac { \sqrt { 5 } }{ \sqrt { 6 } +2 } -\frac { \sqrt { 5 } }{ \sqrt { 6 } -2 } =\frac{\sqrt{5}(\sqrt{6}-2)-\sqrt{5}(\sqrt{6}+2)}{(\sqrt{6}+2)(\sqrt{6}-2)}\)
\(\frac{ ̶​̶​̶​̶​̶​̶​̶​̶√̶3̶0̶-2\sqrt{5}- ̶​̶​̶​̶​̶​̶​̶​̶√̶3̶0̶-2\sqrt{5}}{\sqrt{6}^2-2^2}=\frac{-4\sqrt{5}}{6-4}=\frac{ ̶4̶\sqrt{5}}{ ̶2̶}=-2\sqrt{5}\)

\( \frac{2 \sqrt{6}-\sqrt{5}}{3 \sqrt{5}-2 \sqrt{6}} =\frac{2 \sqrt{6}-\sqrt{5}}{3 \sqrt{5}-2 \sqrt{6}} \times \frac{3 \sqrt{5}+2 \sqrt{6}}{3 \sqrt{5}-2 \sqrt{6}} \)
\( =\frac{6 \sqrt{30}+4 \times 6-3 \times 5-2 \sqrt{30}}{9 \times 5-4 \times 6} \)
\( =\frac{4 \sqrt{30}+24-15}{45-24}=\frac{9+4 \sqrt{30}}{21}\)

\( =\frac{\sqrt{3 \times 16}+\sqrt{2 \times 16}}{\sqrt{3 \times 9}-\sqrt{2 \times 9}} \)
\( =\frac{\sqrt{3 \times 4^{2}}+\sqrt{2 \times 4^{2}}}{\sqrt{3 \times 3^{2}}-\sqrt{2 \times 3^{2}}} \)
\( =\frac{4 \sqrt{3}+4 \sqrt{2}}{3 \sqrt{3}-3 \sqrt{2}} \)
\( =\frac{4 \sqrt{3}+\sqrt{2}}{3 \sqrt{3}-\sqrt{2}}\)
\( =\frac{4(\sqrt{3}+\sqrt{2})}{3(\sqrt{3}-\sqrt{2})} \)
\( =\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}\)
\( =\frac{4}{3}\left(\frac{(\sqrt{3}+\sqrt{2})^{2}}{(\sqrt{3})^{2}-(\sqrt{2})^{2}}\right)\)
\( =\frac{4}{3}\left(\frac{3+2 \sqrt{6}+2}{3-2}\right) \)
\( =\frac{4}{3}\left(\frac{5+2 \sqrt{6}}{1}\right)=\frac{4}{3}(5+2 \sqrt{6}) \)