6.34 \(\times\) 10⁴

\(\sqrt { 63 } -\sqrt { 175 } +\sqrt { 28 } =\sqrt { 9\times 7 } -\sqrt { 25\times 7 } +\sqrt { 4\times 7 } \)
= \(3\sqrt { 7 } -5\sqrt { 7 } +2\sqrt { 7 } \)
= \((3\sqrt { 7 } +2\sqrt { 7 } )-5\sqrt { 7 } \)
= \(5\sqrt { 7 } -5\sqrt { 7 } \)
= 0

\(\sqrt [ 3 ]{ 108 } =\sqrt [ 3 ]{ 27\times 4 } \)
=\(\sqrt [ 3 ]{ { 3 }^{ 3 }\times 4 } \)
=\(\sqrt [ 3 ]{ { 3 }^{ 3 } } \times \sqrt [ 3 ]{ 4 } \) (Laws of radicals - ii)
=\(3\times \sqrt [ 3 ]{ 4 } \) ( Laws of radicals- i)
order = 3; radicand = 4; coefficient = 3

\(\left( \frac { 1 }{ 3 } \right) ^{ -2 }=\frac { 1 }{ \left( \frac { 1 }{ 3 } \right) ^{ 2 } } =\frac { 1 }{ \left( \frac { 1 }{ 9 } \right) } \) = 9

10^-3 = \(\frac { 1 }{ 10^{ 3 } } =\frac { 1 }{ 1000 } \)=0.001

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

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

\(\left( \frac { 1 }{ \sqrt [ 3 ]{ 100 } } \right) ^{ 7 }=(\frac{1}{100^{\frac{1}{3}}})^7=(\frac{1^{\frac{1}{3}}}{100^{\frac{1}{3}}})^7=(\frac{1}{100})^{\frac{7}{3}}\)
\(=10^{\frac{-14}{3}}\)

\((\sqrt[3]{49})^{5}=\left(\sqrt[3]{7^{2}}\right)^{5}=7^{\frac{2 \times 5}{3}}=7^{\frac{10}{3}}\)

\({ 5 }^{ \frac { 1 }{ 2 } }\)