1.6 \(\times\) 10^-1

\((-3)^{4} \times\left(\frac{5}{3}\right)^{4}=(-1 \times 3)^{4} \times \frac{5^{4}}{3^{4}}=(-1)^{4} \times 3^{4} \times \frac{5^{4}}{3^{4}}\)
= (–1)⁴ x 5⁴ = 5⁴ [(–1) ⁴= 1]

(3^-7 \(\div \) 3^-10)\(\times\)3^-5
\(\because\) a^m\(\div \) aⁿ = a^m-n and a^m \(\times\) a ⁿ = a^m+n
\(\therefore\) (3^-7 \(\div \) 3^-10) \(\times\) 3^-5 = [3^-7-(-10)]\(\times\)3^-5
= [3^-7+10]\(\times\) 3⁵ = 3³ \(\times\)3^-5
= 3^3+(-5) = 3^-2=\(\frac { 1 }{ { (3) }^{ 2 } } \)

We have, 0.0000021 =\(\frac{21}{10000000}=\frac{21}{10^{7}}=\frac{2.1\times10}{10^{7}}\)
= \(2.1\times10^{-6}\)                [∵ \(\frac{1}{a^{m}}=a^{-m}\)]
which is the required standard form.

\(\because \quad { \left( \frac { 1 }{ 2 } \right) }^{ -2 }=\frac { 1 }{ { \left( \frac { 1 }{ 2 } \right) }^{ 2 } } =\frac { 1 }{ \left( \frac { 1 }{ 4 } \right) } =1\times \frac { 4 }{ 1 } =4\)
\({ \left( \frac { 1 }{ 3 } \right) }^{ -2 }=\frac { 1 }{ { \left( \frac { 1 }{ 3 } \right) }^{ 2 } } =\frac { 1 }{ \left( \frac { 1 }{ 9 } \right) } =1\times \frac { 9 }{ 1 } =9\)
\({ \left( \frac { 1 }{ 4 } \right) }^{ -2 }=\frac { 1 }{ { \left( \frac { 1 }{ 4 } \right) }^{ 2 } } =\frac { 1 }{ \left( \frac { 1 }{ 16 } \right) } =1\times \frac { 16 }{ 1 } =16\)
\(\therefore \quad { \left( \frac { 1 }{ 2 } \right) }^{ -2 }+{ \left( \frac { 1 }{ 3 } \right) }^{ -2 }+{ \left( \frac { 1 }{ 4 } \right) }^{ -2 }\)
= 4 + 9 + 16 = 29

We have \((-4)^{5}\div(-4)^{8}\)
=\(\frac{(-4)^{5}}{(-4)^{8}}=\frac{1}{(-4)^{8}\times (-4)^{-5}}\)    \([\because a^{m}=\frac{1}{a^{-m}}]\)
=\(\frac{1}{(-4)^{8-5}}=\frac{1}{(-4)^{3}}\)     [∵ a^mxaⁿ=a^mxn]
which is the required form.

We have, \(P^{3}\times P^{-10}\)=\(P^{3}\times \frac{1}{P^{10}}\)    [\(a^{-m}=\frac{1}{a^{m}}\)]
=\(\frac{p^{3}}{p^{10}}=\frac{1}{p^{10}p^{-3}}=\frac{1}{p^{10-3}}\)         [∵ a^m.aⁿ=a^m+n]
=\(\frac{1}{p^{7}}=p^{-7}\)        [∵ \(a^{-m}=\frac{1}{a^{m}}\)]
which is the required exponential form.

We have, \((-2)^{-3}\times (-2)^{-4}=\frac{1}{(-2)^{3}}\times \frac{1}{(-2)^{4}}\)
[ஃ  a^m \(\times\) aⁿ = a^m+n, where m and n are integers]
\(=\frac{1}{(-2)^{3+4}}=\frac{1}{(-2)^{7}}=(-2)^{-7}\)
which is the required exponential form.

∵ \(\frac{6^{n}}{6^{-2}}=6^{3}\)
∴ 6ⁿ⁺²=6³ ⇒ n+2=3 ⇒ n=3-2
⇒ n=1  [∵ bases of two equal numbers are same, so exponents will be same]

1.25x 10^-4

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

We have, 144 x 2^-3 = \(\frac{144}{2^{3}}=\frac{12\times 12}{2\times2\times 2 }=18\)
Now, 18X12^-1=\(\frac{18}{12}=\frac{3}{2}\)
Again, \(\frac{3}{2}\times 3^{-2}=\frac{3}{2}\times \frac{1}{3^{2}}=\frac{1}{6}\)
So,
 

(a) Earth ➝ 1.496 x 10⁸ km
Jupiter ➝ 7.783 x 10⁸ km
Mars ➝ 2.279 x 10⁸ km
Mercury ➝ 5.79 x 10⁷ km
Neptune ➝ 4.497 x 10⁹ km
Pluto ➝ 5.9 x 10⁹ km
Saturn ➝ 1.427 x 10⁹ km
Uranus ➝ 2.87 x 10⁹ km
Venus ➝ 1.082 x 10⁸ km
(b) On arranging the planets from closest to the Sun to farthest from the Sun, we get following order
Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto.

783000

4.42 X10²