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}
(a^m) x (aⁿ) = a^{m + n}
a^m ÷ aⁿ = a^{m ? n}
and if p^X = p^Y then X will be equal to Y. means X = Y;

Correct Option: A

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

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

Solve the equation by using the Algebra Law.

Correct Option: D

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

Given equation is
2^p + 3^q = 17
2^{p + 2} - 3^{q + 1} = 5

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}

Correct Option: C

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

^L√3 = M^1/L = Surd of L^th order.

Correct Option: A

^L√3 = M^1/L = Surd of L^th order.

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

Correct Option: C

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 .