^L?M is surd of order ........, where M is a rational number; L is a positive integer and ^L?M is irrational.

Given that , 289 = 17^x/5
Use the formula if a^x = a ^y then x = y will be true.
? 17^1/2 = 17^x/5
? x /5 = 1/2
? x = 10 .

17^3.5 x 17^7.3 ÷ 17^4.2 = 17^?
Apply the Laws of Exponents ::
(a^m) x (aⁿ) = a^m+n
Fractional Exponents ::
a^m÷ a ⁿ=a^m?n
? 17^{3.5 + 7.3 - 4.2} = 17^?
? 17^6.6 = 17^?
? ? = 6.6

Third power of 4 = 4³ = 4 × 4 × 4 = 64

a⁵ × a⁷ = a^{5 + 7} = a¹²

Apply the law of Fractional Exponents and Laws of Exponents
(a^m) x (aⁿ) = a^{m + n}
a^-m = 1/a^m
p⁰ = 1
and if p^X = p^Y then X will be equal to Y. means X = Y;
(a^m)ⁿ = a^{m x n}
(a^m) x (aⁿ) = a^{m + n}
a^m ÷ aⁿ = a^{m ? n}
Or
a^m/aⁿ = a^{m ? n}

Given equation is
2^p + 3^q = 17 ----------------(i)
2^{p + 2} - 3^{q + 1} = 5
? 2^p x 2² - 3^q x 3 = 5
? 2^p x 4 - 3q x 3 = 5
? 4 x 2^p - 3 x 3q = 5---------------(ii)
On multiplying Eq. (i) by 3 and adding it, with Eq. (ii), we get

3 x 2^p + 3 x 3^q = 51
4 x 2^p - 3 x 3^q = 5
_____________________________
7 x 2^p = 56

? 7 x 2^p = 7 x 8
? 2^p = 8
? 2^p = 2³
? p = 3

On substituting the value of p in Eq. (i) . we get
2³ + 3^q = 17
? 8 + 3^q = 17
? 3^q = 17 - 8
? 3^q = 9
? 3^q = 3²
? q = 2

Given that, (42 x 229) ÷ (9261)^1/3 = ?
As we know that (9621)^1/3 = (21 x 21 x 21)^1/3 = 21^3x1/3
? 42 x 229 ÷ (21^{3 x} )^1/3 = ?
? 42 x 229 ÷ 21 = ?
? 42 x 229 x 1/ 21 = ?
? 2 x 229 = ?
? ? = 2 x 229
? ? = 458

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