\(\sqrt [ 3 ]{ 40 } \times \sqrt [ 3 ]{ 16 } =\left( \sqrt [ 3 ]{ 2\times 2\times 2\times 5 } \right) \times \left( \sqrt [ 3 ]{ 2\times 2\times 2\times 2 } \right) \)
=\(\left( 2\times \sqrt [ 3 ]{ 5 } \right) \times \left( 2\times \sqrt [ 3 ]{ 2 } \right) =4\times \left( \sqrt [ 3 ]{ 2 } \times \sqrt [ 3 ]{ 5 } \right) =4\times \sqrt [ 3 ]{ 2\times 5 } \)
= \(4\sqrt [ 3 ]{ 10 } \) .

\(\sqrt [ 3 ]{ 192 } =\sqrt [ 3 ]{ 4\times 4\times 4\times 3 } =4\sqrt [ 3 ]{ 3 } \)

\(\sqrt { 8 } =\sqrt { 4\times 2 } =2\sqrt { 2 } \)

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

(i) 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

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

\(\left( \frac { 64 }{ 125 } \right) ^{ \frac { -2 }{ 3 } }=(\frac{125}{64})^{\frac{2}{3}}=((\frac{5}{4})^3)^{\frac{2}{3}}=(\frac{5}{4})^{ ̶3̶\times\frac{2}{ ̶3̶}}=(\frac{5}{4})^2=\frac{25}{16}\)

\(\left( 49 \right) ^{ \frac { 1 }{ 2 } }=(7\times7)^{\frac{1}{2}}=7\)

32 = 4\(\times\) 4\(\times\)\(4^{\frac{1}{2}}\)
=\(4^{2+\frac{1}{2}}={ 4 }^{ \frac { 5 }{ 2 } }\)

\(\frac{1}{5}\) = 5^-1