30⁶

(a) 1x10¹⁰⁰
(b) 10²⁰⁰

(i) 3.1865 x 10¹⁰
(ii) 3.90 x 10³

(i) \(\frac { 1 }{ 256 } \)
(ii) \(-\frac { 8 }{ 27 } \)

Distance between Earth and Moon = 384000000 m
=384 x 10⁶             
=3.84 x10⁸m                    ........(i)
and Distance between Sun and Earth = 146900000000
= 1469x10⁸                     .......(ii)
Comparing Eqs. (i) and (ii), we see that
3.84 x10⁸ < 1469 x10⁸    [\(\because\) 3.84 < 1469]
So, distance between Sun and Earth is more than distance between Earth and Moon.

Given, \(\frac { \left( -\frac { 3 }{ 4 } \right) ^{ 4 }\times \left( \frac { 125 }{ 27 } \right) }{ \left( \frac { 5 }{ 3 } \right) ^{ 2 }\times \left( \frac { 9 }{ 16 } \right) } \)
∵ \(\frac { 125 }{ 27 } =\frac { 5\times 5\times 5 }{ 3\times 3\times 3 } =\frac { 5^{ 3 } }{ 3^{ 3 } } \)
and \(\frac { 9 }{ 16 } =\frac { (-3)\times (-3) }{ 4\times 4 } =\frac { (-3)^{ 2 } }{ { 4 }^{ 2 } } \)
So, \(\frac { \left( -\frac { 3 }{ 4 } \right) ^{ 4 }\times \frac { { 5 }^{ 3 } }{ { 3 }^{ 3 } } }{ \left( \frac { 5 }{ 3 } \right) ^{ 2 }\times \frac { (-3)^{ 2 } }{ { 4 }^{ 2 } } } =\frac { \left( -\frac { 3 }{ 4 } \right) ^{ 4 }\times \left( \frac { 5 }{ 3 } \right) ^{ 3 } }{ \left( \frac { 5 }{ 3 } \right) ^{ 2 }\times \left( -\frac { 3 }{ 4 } \right) ^{ 2 } } \)      \(\left[ \because \frac { { a }^{ n } }{ b^{ n } } =\left( \frac { a }{ b } \right) ^{ n } \right] \)
= \(\left( \frac { -3 }{ 4 } \right) ^{ 4-2 }\times \left( \frac { 5 }{ 3 } \right) ^{ 3-2 }\)     [∵ a^m ÷ aⁿ = a^m-n]
=\(\left( \frac { -3 }{ 4 } \right) ^{ 2 }\times \left( \frac { 5 }{ 3 } \right) ^{ 1 }\)
= \(\frac { (-3)\times (-3) }{ 4\times 4 } \times \frac { 5 }{ 3 } =\frac { 9 }{ 16 } \times \frac { 5 }{ 3 } =\frac { 3\times 5 }{ 16 } =\frac { 15 }{ 16 } \).

Given, \({ \left( \frac { 1 }{ 5 } \right) }^{ 5 }\times { \left( \frac { 1 }{ 5 } \right) }^{ 19 }={ \left( \frac { 1 }{ 5 } \right) }^{ 8x }\)
\(\Rightarrow ​​{ \left( \frac { 1 }{ 5 } \right) }^{ 5+19 }={ \left( \frac { 1 }{ 5 } \right) }^{ 8x }\)      [\(\because\)a^m x aⁿ = a^m+n]
\(\Rightarrow ​​{ \left( \frac { 1 }{ 5 } \right) }^{ 24 }={ \left( \frac { 1 }{ 5 } \right) }^{ 8x }\)
Since, bases are equal, by equating their exponents, we get
8x = 24
\(\therefore\)  x = 24/8 = 3

\(\frac { { 5 }^{ -2 }\times { 3 }^{ -3 }\times (125)^{ 2/3 } }{ (27)^{ -2/3 }\times (32)^{ -1/5 } } \)
∵ 125 = (5)³ = 5 x 5 x 5
So, (125)^2/3 =(5)^{3 x 2/3} =5² and 27=(3)³
∴ (27)^-2/3 = {(3)³}^-2/3 =\({ (3 })^{ 3\times \frac { -2 }{ 3 } }\) =(3)²
32 = 2 x 2 x 2 x 2 x 2 = (2)⁵
So, (32)^-1/5 = {(2)⁵}^-1/5 = \({ (2) }^{ 5\times \frac { (-1) }{ 5 } }\)=(2)^-1
Now \(\frac { { 5 }^{ -2 }\times { 3 }^{ -3 }\times { 5 }^{ 2 } }{ (3)^{ -2 }\times (2)^{ -1 } } \)     \(\left[ \because a^{ -m }=\frac { 1 }{ { a }^{ m } } \right] \)
= 5^-2 x 3^-3 x 3² x 2¹ x 5²
=5^-2+2 x 3^-3+2 x 2¹
a^m x aⁿ =a^m+n
=5⁰ x 3^-1 x 2¹ = 1 x \(\frac { 1 }{ 3 } \times 2=\frac { 2 }{ 3 } \).

Given, 1.75 x1 0^-3
1.75 = 175 x 10^-2
So, 1.75 x 10^-3 =175 x 10^-3 x 10^-2
=175 x 10^-5          \(\left[ \because \frac { 1 }{ { 10 }^{ 5 } } ={ 10 }^{ -5 } \right] \)
= 0.00175

Five such examples are given below:
(i) We have, 4096

= 4x4x4x4x4x4 = 4⁶
Thus, the exponential form of 4096 is 4⁶.
Here, base = 4 and exponent = 6
(ii) We have, 216

=2x2x2x3x3x3
= 2³ x 3³ = (2 x 3)³ = 6³
Thus, exponential form of 216 is 6³
Here, base = 6 and exponent = 3
(iii) We have, 15625

= 5 x 5 x 5 x 5 x 5 x 5 = 5⁶
Thus, exponential form of 15625 is 5⁶.
Here, base = 5 and exponent = 6
(iv) We have, 1331

= 11 x 11 x 11 = 11³
Thus, exponential form of 1331 is 11³
Here, base = 11 and exponent = 3
(v) We have, 196

= 2 x 2 x 7 x 7 = 2² X 7² = (2 x 7)² = 14²
Thus, exponential form of 196 is 14².
Hence, base = 14 and exponent = 2