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^?

Correct Option: C

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

Given equation is,
( 6a^-2bc^-3/4ab^-3c² ) ÷ ( 5a^-3 b²c^-1/3ab^-2c³ )
Apply the algebra law a ÷ b = a x 1/b
Apply the law of Fractional Exponents and Laws of Exponents
(a^m)(a^{n/) = am+n
am÷an=am-n
Or
am/an=am-n
Solve the equation.}

Correct Option: B

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]

Given expression = [(√2)^√2]^√2
= (√2)^(2)√2/2
Apply the law of algebra formula and solve the equation.

Correct Option: D

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,
( 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)

Correct Option: A

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

√5 + ∛x = 3
Apply the law of Algebra Law.
Square both side of equation and solve the equation.

Correct Option: A

√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