∵ 3^x - 3^{x - 1} = 18
⇒ 3^{x - 1}(3 - 1) = 18
Apply the Algebra law,
If a^X = a^Y then X will be equal to Y.
means X = Y;

Correct Option: C

∵ 3^x - 3^{x - 1} = 18
⇒ 3^{x - 1}(3 - 1) = 18
⇒ 3^{x - 1}(2) = 18
⇒ 3^{x - 1} = 18/2
⇒ 3^{x - 1} = 9
⇒ 3^{x - 1} = 3²
Apply the Algebra law,
If a^X = a^Y then X will be equal to Y.
means X = Y;
⇒ x - 1 = 2
⇒ x = 3
Then x^x = (3)³ = 27

∵ 2^{x - 1} + 2^{x + 1} = 320
Apply the law of Algebra
⇒ 2^{x - 1}(1 + 2 ² ) = 320
Solve the equation.

Correct Option: C

∵ 2^{x - 1} + 2^{x + 1} = 320
Apply the law of Algebra
⇒ 2^{x - 1}(1 + 2 ² ) = 320
⇒ 2^{x - 1}(1 + 4 ) = 320
⇒ 2^{x - 1} x 5 = 320
⇒ 2^{x - 1} = 64 = 2 x 2 x 2 x 2 x 2 x 2
⇒ 2^{x - 1} = 2⁶
if p^X = p^Y then X will be equal to Y. means X = Y;
⇒ x - 1 = 6
∴ x = 7

√5 + 2√6 - 1/√5 - 2√6
= ((√5 + 2√6 x √5 - 2√6 ) - 1)/√5 - 2√6

Apply the formula of Algebra.
( a + b ) x (a - b ) = a² - b²

Correct Option: B

√5 + 2√6 - 1/√5 - 2√6
= ((√5 + 2√6 x √5 - 2√6 ) - 1)/√5 - 2√6

Apply the formula of Algebra.
( a + b ) x (a - b ) = a² - b²
= (√(5)² - (2√6 )² - 1)/√5 - 2√6
= (√25 - 4 x 6 - 1) /√5 - 2√6
= (√25 - 24 - 1) /√5 - 2√6
= (√ 1 - 1 )/√5 - 2√6
= (1 - 1 )/√5 - 2√6
= 0/√5 - 2√6
= 0

Given equation is
(10)²⁰⁰ ÷ (10)¹⁹⁶
Apply the law of Algebra
a^m ÷ aⁿ = a^{m ? n}
= (10)^{200 - 196}

Correct Option: A

Given equation is
(10)²⁰⁰ ÷ (10)¹⁹⁶
Apply the law of Algebra
a^m ÷ aⁿ = a^{m ? n}
= (10)^{200 - 196}
= 10⁴
= 10000

2^{x - 1} + 2^{x + 1} = 2560
2^{x - 1} + 2^{x - 1 + 2} = 2560
Apply the Law of Algebra
Solve the equation.

Correct Option: A

2^{x - 1} + 2^{x + 1} = 2560
2^{x - 1} + 2^{x - 1 + 2} = 2560
Apply the Law of Algebra
2^{x - 1} + 2^{x - 1} x 2 ² = 2560
⇒ 2^{x - 1} ( 1 + 2² ) = 2560
⇒ 2^{x - 1} = 2560/5 = 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
⇒ 2^{x - 1} = 2⁹
⇒ x - 1 = 9
∴ x = 9 + 1 = 10