(i) \(\sqrt [ 3 ]{ 5 } ,\sqrt [ 9 ]{ 4 } ,\sqrt [ 6 ]{ 3 } \)
\(5^{\frac{1}{3}}\)
ஃ The order of the surds \(\sqrt [ 3 ]{ 5 } ,\sqrt [ 9 ]{ 4 } ,\sqrt [ 6 ]{ 3 } \) are 3,9,6
\(4^{\frac{1}{9}}\)
\(3^{\frac{1}{6}}\) l.c.m of 3,9,6 is 18
ஃ \(\frac{1}{3}=\frac{1\times6}{3\times6}=\frac{6}{18}\)
\(\frac{1}{9}=\frac{1\times2}{9\times2}=\frac{2}{18};\frac{1}{6}=\frac{1\times3}{6\times3}=\frac{3}{18}\)
\((5^{\frac{1}{3}})=5^{\frac{6}{18}}=(15625)^{\frac{1}{18}}\)
\((4^{\frac{1}{9}})=4^{\frac{2}{18}}=(4^2)^{\frac{1}{18}}=16^{\frac{1}{18}}\)
\((3^{\frac{1}{6}})=3^{\frac{3}{18}}=(3^3)^{\frac{1}{18}}=27^{\frac{1}{18}}\)
ஃ The descending order of \(\sqrt [ 3 ]{ 5 } ,\sqrt [ 9 ]{ 4 } ,\sqrt [ 6 ]{ 3 } \) is \((15625)^{\frac{1}{18}}>(27)^{\frac{1}{18}}>16^{\frac{1}{18}}\) i.e., \(\sqrt { \sqrt { 3 } } >\sqrt [ 2 ]{ \sqrt [ 3 ]{ 5 } } >\sqrt [ 3 ]{ \sqrt [ 4 ]{ 7 } } \)
(ii) \(\sqrt [ 2 ]{ \sqrt [ 3 ]{ 5 } } ,\sqrt [ 2 ]{ \sqrt [ 3 ]{ 5 } } ,\sqrt [ 3 ]{ \sqrt [ 4 ]{ 7 } } ,\sqrt { \sqrt { 3 } } \)
The order of the surds \(\sqrt [ 2 ]{ \sqrt [ 3 ]{ 5 } } ,\sqrt [ 2 ]{ \sqrt [ 3 ]{ 5 } } ,\sqrt [ 3 ]{ \sqrt [ 4 ]{ 7 } } ,\sqrt[2] { \sqrt [2]{ 3 } } \)  are 6, 12, 4
l.c.m of 6,12,4 is 12
\(\sqrt[2]{\sqrt[3]{5}}=5^{\frac{1}{6}}=5^{\frac{1\times2}{6\times2}}=5^{\frac{2}{12}}=(5^2)^{\frac{1}{12}}=25^{\frac{1}{12}}\)
\(\sqrt[3]{\sqrt[4]{\sqrt{7}}}=7^{\frac{1}{12}};\sqrt{\sqrt{3}}=3^{\frac{1}{4}}=3^{\frac{1\times3}{4\times3}}=3^{\frac{3}{12}}=(3^3)^{\frac{1}{12}}=27^{\frac{1}{12}}\)
ஃ The ascending order of the surds
\(\sqrt [ 2 ]{ \sqrt [ 3 ]{ 5 } } ,\sqrt [ 2 ]{ \sqrt [ 3 ]{ 5 } } ,\sqrt [ 3 ]{ \sqrt [ 4 ]{ 7 } } ,\sqrt { \sqrt { 3 } } \) is \(7^{\frac{1}{12}}<25^{\frac{1}{2}}<27^{\frac{1}{2}}\), that is \(\sqrt[3]{\sqrt[4]{7}}<\sqrt[2]{\sqrt[3]{5}}<\sqrt{\sqrt{3}}\)

(i) \(\sqrt{3}\times\sqrt{5}\times\sqrt{2}=\sqrt{3\times5\times2}=\sqrt{30}\)
(ii) \(\sqrt{35}\div\sqrt{7}=\sqrt{\frac{35}{7}}=\sqrt{5}\)
(iii) \(\sqrt[3]{27}\times\sqrt[3]{8}\times\sqrt[3]{125}=\sqrt[3]{27\times8\times125}=\sqrt[3]{3^3\times2^3\times5^3}=3\times2\times5=30\)
(iv) \((7\sqrt{a}-5\sqrt{b})(7\sqrt{a}+5\sqrt{b})=(7\sqrt{a})^2-(5\sqrt{b})^2=49a-25b\) = 21\(\sqrt 3\)
(v) \([\sqrt{\frac{225}{729}}-\sqrt{\frac{25}{144}}]\div\sqrt{\frac{16}{81}}\)

=\([\sqrt{\frac{15^2}{27^2}}-\sqrt{\frac{5^2}{12^2}}]\times\sqrt{\frac{9^2}{4^2}}\)
=\((\frac{15}{27}-\frac{5}{12})\times\frac{9}{4}=(\frac{5}{9}-\frac{5}{12})\times\frac{9}{4}\)
=\((\frac{20-15}{36})\times\frac{9}{4}=\frac{5}{ ̶3̶6̶}\times\frac{ ̶9̶}{4}=\frac{5}{16}\)

(i) 3\(\sqrt{7}\) + 5\(\sqrt{7}\) = (3 + 5)\(\sqrt{7}\) = 8\(\sqrt{7}\). The answer is irrational.
(ii) 7\(\sqrt{5}\) - 4\(\sqrt{5}\) = (7 - 4)\(\sqrt{5}\) = 3\(\sqrt{5}\). The answer is irrational.

(i) The world population is nearly
7,000,000,000 = 7.0 \(\times\) 10⁹
(ii) One light year means the distance
9460528400000000 km = 9.4605284 \(\times\) 10¹⁵ km
(iii) Mass of an electron is
0.000 000 000 000 000 000 000 000 000 00091093822 kg = 9.1093822 \(\times\) 10^-31 kg

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

(4000000)³ ÷ (0.00002)⁴
\( =\left(4.0 \times 10^{6}\right)^{3} \div\left(2.0 \times 10^{-5}\right) 4 \)
\(=(4.0)^{3} \times\left(10^{6}\right)^{3} \div(2.0)^{4} \times\left(10^{-5}\right)^{4} \)
\(=\frac{64.0 \times 10^{18}}{16.0 \times 10^{-20}} \)
\(=4 \times 10^{18} \times 10^{+20} \)
\(=4.0 \times 10^{38} \)

(300000)³ \(\times\) (2000)⁴
\( =\left(3.0 \times 10^{5}\right)^{3} \times\left(2.0 \times 10^{3}\right)^{4} \)
\( =(3.0)^{3} \times\left(10^{5}\right)^{3} \times(2.0)^{4} \times\left(10^{3}\right)^{4} \)
\( =(27.0) \times\left(10^{15}\right) \times(16.0) \times\left(10^{12}\right) \)
\(= \left(2.7 \times 10^{1}\right) \times\left(10^{15}\right) \times\left(1.6 \times 10^{1}\right) \times\left(10^{12}\right) \)
\( =2.7 \times 1.6 \times 10^{1} \times 10^{15} \times 10^{1} \times 10^{12} \)
\(=4.32 \times 10^{1+15+1+12}=4.32 \times 10^{29} \)

Total mass = 5.97\(\times\) 10²⁴ kg + 0.073 \(\times\) 10²⁴ kg
= (5.97 + 0.073) \(\times\) 10²⁴ kg
= 6.043 \(\times\) 10²⁴ kg