81^2.5 X 9^4.5 ÷ 3^4.8 = 9^?
? 81^2.5 x 9^4.5 ÷ 3^4.8 = 9^?
Apply the law of Fractional Exponents and Laws of Exponents
(a^m)ⁿ = a^{m x n} ----------------1
(a^m) x (aⁿ) = a^{m + n}-------------2
a^m ÷ aⁿ = a^{m ? n}----------------3
and if p^X = p^Y then X will be equal to Y. means X = Y;----------------4

Apply Law 1
? (9²)^2.5 x 9^4.5 ÷ 3^{2 x 2.4} = 9^?
? 9⁵ x 9^4.5 ÷ 9^2.4 = 9^?
Apply law 2 and 3,
? 9^{(5 + 4.5 - 2.4)} = 9^?
? 9^? = 9^7.1
Apply the Law 4
? ? = 7.1

?5 + ?x = 3
Apply the law of Algebra Law.
Square both side of equation.
(?5 + ?x)² = 3²
? 5 + ?x = 9
? ?x = 9 - 5
? ?x = 4
Now cube the both side of equation,
? ( ?x ) ³ = 4³
? x = (4)³
? x = 64

Given equation is,
( 5 + 2?3 ) / (7 + 4?3 )
Now Multiply and divide by ( 7 - 4?3 ) in above given equation.
By Rationalization, we will get,
[ ( 5 + 2?3 ) / ( 7 + 4?3 ) ] x [ ( 7 - 4?3 ) / ( 7 - 4?3 ) ]
[ ( 5 + 2?3 ) x ( 7 - 4?3 ) ] / [ ( 7 + 4?3 ) x ( 7 - 4?3 ) ]
Apply the formula and multiplication rule of algebra,
( P + Q ) x (R - S ) = P x R - P x S + Q x R - Q x S .................(1)
( a + b ) x ( a - b) = a² - b².......................................................(2)
= ( 5 x 7 - 5 x 4?3 + 7 x 2?3 - 2?3 x 4?3 )/ ( 7² - (4?3)² )
= 35 - 20?3 + 14?3 - 8 x 3 / ( 49 - 16 x 3 )
= ( 35 - 24 - 6?3 )/(49 - 48 )
= 11 - 6?3
? ( 5 + 2?3 ) /( 7 + 4?3 ) = a + b?3
? a + b?3 = 11 - 6?3
on equating the above equation and we get,
or a = 11 and b = - 6

Given expression = [(?2)^?2]^?2
= (?2)^(2)?2/2
= (?2)^(2)1/?2
= (2)^1/2 x 2^1/?2
= 2^(2/21/?2)
= (2)^{(2)(1/?2 - 1)}
which denotes a real number but not a rational number .

Given equation is,
( 6a^-2bc^-3/4ab^-3c² ) ÷ ( 5a^-3 b²c^-1/3ab^-2c³ )
Apply the algebra law a ÷ b = a x 1/b
= ( 6a^-2bc^-3/4ab^-3c² ) x ( 3ab^-2c³/5a^-3 b²c^-1 )
= ( 6a^-2 bc^-3 x 3ab^-2c³) / ( 4ab^-3c² x 5a^-3b²c^-1 )
Apply the law of Fractional Exponents and Laws of Exponents
(a^m)(aⁿ) = a^m+n
a^m÷aⁿ=a^m-n
Or
a^m/aⁿ=a^m-n
= 18a^{-2 + 1} b^{1 - 2} c ^{-3 + 3} / 20a^{1 - 3} b^{-3 + 2} c^{2 - 1}
= 9a^-1b^-1c⁰/ 10a ^{- 2} b^-1c¹
= ( 9/10 ) a^{-1 + 2}b^{-1 + 1} c^{0 - 1}
= ( 9/10 ) a¹b⁰ c^-1
= ( 9/10 ) ac^-1 [? b⁰ = 1]

? 10^2/5 × 10^8/5 = 10ⁿ
? 10^{2/5 + 8/5} = 10ⁿ
? 10^10/5 = 10ⁿ ? 10² = 10ⁿ ? n = 2
Hence, the value of n is 2.