100

32

2³ x 3³

3²>2³

NO

we have \([{5^6\over 5^3}]\times 5^2\)  =[5^6-3]x5²      \((\because {a^m\over a^n}=a^{m-n})\)
= 5³ X 5² 
= 5^{3 + 2}                                         \((\because a^m \times a^n =a^{m+n})\)
= 5⁵
Thus,\([{5^6\over 5^3}]\times 5^2=5^5\)

1. x \(\times\) x \(\times\) x \(\times\)x = x⁴ is read as 'x raised to the power 4' or '4^th power of x'.
2.x⁵y⁵ is read as 'x squared into y raised to power 5'.
3. p⁶q³ is read as 'p raised to the power 6 into q cubed'.

\(\frac{10\times 5^{n+1}+25\times 5^n}{3\times 5^{n+2}+10\times 5^{n+1}}\)
\(=\frac{2\times 5\times 5^n\times 5^1+5\times5\times5^n}{3\times5^n\times5^2+2\times5\times5^1\times5^n}\)  
\(=\frac{2\times5\times5^n\times5+5\times5\times5^n}{3\times5^n\times5\times5+2\times5\times5\times5^n}\)
\(=\frac{5\times5\times5^n\times2+5\times5\times5^n}{5\times5\times5^n\times3+5\times5\times5^n\times2}\)
\(=\frac{5\times5\times5^n(2+1)}{5\times5\times5^n(3+2)}=\frac{(2+1)}{(3+2)}\)
\(=\frac{3}{5}\)

45342

b²

169

Given, 2ⁿ⁺²- 2ⁿ = c x 2ⁿ
⇒ 2ⁿ⁺² = 2ⁿ x 2², 2ⁿ⁺¹ = 2ⁿ x 2¹
So, 2ⁿ x 2² - 2ⁿ x 2¹ + 2ⁿ = c x 2ⁿ
Taking ~ common from both the side, we get
2ⁿ (2²-2¹)= c x 2ⁿ
⇒ (2²-2¹+1)c
∴ c = 4-2+1   [∵ 2²= 2 x 2=4, 2¹=2]
=2 +1 =3

Given, \({ 2 }^{ n-5 }\times { 6 }^{ 2n-4 }=\frac { 1 }{ 12^{ 4 }\times 2 } \)
∵ 6^2n-4 = (2 x 3)^2n-4 = 2^2n-4 x 3^2n-4
and 12⁴ = (3 x 4)⁴ =(3 x 2 x 2)⁴ = 3⁴ x 2⁴ x 2⁴
So, 2^n-5 x 2^2n-4 x 3^2n-4 =\(\frac { 1 }{ { 3 }^{ 4 }\times { 2 }^{ 4 }\times { 2 }^{ 4 }\times { 2 }^{ 1 } } \)
 ⇒ 2^n-5+2n-4 x 3^2n-4 =\(\frac { 1 }{ { 3 }^{ 4 }\times { 2 }^{ 4+4+1 } } \)
[∵ a^m x aⁿ =a^m+n]
⇒ 2^3n-9 x 3^2n-4 = \(\frac { 1 }{ { 3 }^{ 4 }\times { 2 }^{ 9 } } \)
⇒ 2^3n-9 x 3^2n-4 = 3^-4 x 2^-9
[∵ a^-m=\(\frac { 1 }{ { a }^{ m } } \)]
a^m =aⁿ ⇒  m=n
So,3n-9 = -9
3n = - 9 + 9 = 0 ⇒ n = 0

Given, 1029
∵ 1029 = 3 x 7 x 7 x 7 = 3 x 7³

The exponent form of 1029 is 3 x 7³.

Given, 1024
∵ 1024 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2¹⁰

The exponent form of 1024 is 2¹⁰.