Here, Let us Assume ,
p = √56 + √56 + √56 + ...........
⇒ p = √56 + p
Square on both side and solve the above equation.

Correct Option: C

Here, Let us Assume ,
p = √56 + √56 + √56 + ...........
⇒ p = √56 + p
Square on both side and solve the above equation.
⇒ p²= 56 + p
⇒ p² - p= 56
⇒ p(p - 1 ) = 56
⇒ p(p - 1 ) = 8 x 7
⇒ Either p = 8 or p - 1 = 7
⇒ So p = 8;
Now given equation in question is
√56 + √56 + √56 + .... ) ÷ 2² = ?
⇒ p ÷ 2² = ?
Put the value of p
⇒ 8 ÷ 2² = ?
⇒ 8 ÷ 4 = ?
⇒ ? = 2;

( 9ⁿ x 3² x (3^-n/2)^-2 - (27)ⁿ ) / ( 3^3m x 2³ ) = 1/27
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

⇒ ( 3²ⁿ x 3² x 3ⁿ - 3³ⁿ ) / ( 3^3m x 2³ ) = 1/27
Solve the equation.

Correct Option: A

( 9ⁿ x 3² x (3^-n/2)^-2 - (27)ⁿ ) / ( 3^3m x 2³ ) = 1/27
⇒ ( 3²ⁿ x 3² x 3ⁿ - 3³ⁿ) / ( 3^3m x 2³ ) = 1/27
⇒ ( 3^{2n + n} x 3² - 3³ⁿ) / ( 3^3m x 2³ )= 1/27
⇒ ( 3³ⁿ x 3² - 3³ⁿ) / ( 3^3m x 2³ ) = 1/27
⇒ 3³ⁿ( 3² - 1) / ( 3^3m x 8 ) = 1/27
⇒ 3³ⁿ( 9 - 1) / (3^3m x 8 ) = 1/27
⇒ ( 3³ⁿ x 8 ) / (3^3m x 8 ) = 1/27
⇒ 3³ⁿ / 3^3m = 1/27
⇒ 3^{3n - 3m} = 1/3³
⇒ 3^{3( n - m )} = 3^-3
3(n - m) = -3
or n - m = -1
or m - n = 1

You can square the both the equation and add them to find the answer.

x² + y² = ( (√3 + 1 ) / ( √3 - 1 ) )² + ( (√3 - 1 ) / (√3 + 1) )²
x² + y² = ( √3 + 1 )² / ( √3 - 1 ) ² + (√3 - 1 )² / (√3 + 1) ²

Solve further by applying the Law of Algebra formula

Correct Option: A

You can square the both the equation and add them to find the answer.

x² + y² = ( (√3 + 1 ) / ( √3 - 1 ) )² + ( (√3 - 1 ) / (√3 + 1) )²
x² + y² = ( √3 + 1 )² / ( √3 - 1 ) ² + (√3 - 1 )² / (√3 + 1) ²
x² + y² = ( ( 3 + 1 + 2 √3 ) / ( 3 + 1 - 2 √3 ) ) + ( ( 3 + 1 - 2√3) / ( 3 + 1 + 2 √3 ) )
x² + y² = ( 4 + 2 √3 ) / ( 4 - 2 √3 ) + ( 4 - 2√3 ) / ( 4 + 2 √3 )
x² + y² = ( 2 + √3 ) / ( 2 - √3 ) + ( 2 - √3 ) / ( 2 + √3 )
x² + y² = (4 + 3 + 4 √3 + 4 + 3 - 4 √3) / ( 4 - 3 )
x² + y² = 14

Given equation is
a^-1 - 1
Apply the law of Fractional Exponents and Laws of Exponents
x^-y = 1/x^y
Rationalize the above equation and divide as per given question and solve it.

Correct Option: D

Given equation is
a^-1 - 1
Apply the law of Fractional Exponents and Laws of Exponents
x^-y = 1/x^y
⇒ a^-1 - 1 = 1/a - 1
⇒ a^-1 - 1 = (1 - a)/a
Now divide by a - 1 in above equation,
⇒ ( a^-1 - 1 ) ÷ (a - 1) = (1 - a)/a ÷ (a - 1)
⇒ ( a^-1 - 1 ) ÷ (a - 1) = (1 - a)/a x 1/ (a - 1)
⇒ ( a^-1 - 1 ) ÷ (a - 1) = -1x (a - 1)/a x 1/ (a - 1)
⇒ ( a^-1 - 1 ) ÷ (a - 1) = -1/a
∴ Required quotient = -1/a.

Given that 16 x 8^{n + 2} = 2^m

Apply the law of Fractional Exponents and Laws of Exponents
if a multiply three times a x a x a then
a x a x a = a³
if a multiply two times a x a then
a x a = a²
if a multiply n times a x a x a x....up to n times, then
a x a x a x a ......up to n times = aⁿ
and if p^X = p^Y then X will be equal to Y. means X = Y;
Law of exponent
a^maⁿ = a^m+n

Correct Option: D

Given that 16 x 8^{n + 2} = 2^m

Apply the law of Fractional Exponents and Laws of Exponents
if a multiply three times a x a x a then
a x a x a = a³
if a multiply two times a x a then
a x a = a²
if a multiply n times a x a x a x....up to n times, then
a x a x a x a ......up to n times = aⁿ
⇒ (2)⁴ x 2^{3 x (n+2)} = 2^m
⇒ (2)⁴ x 2³ⁿ⁺⁶ = 2^m
a^maⁿ = a^m+n
⇒ (2)^{(4 + 3n + 6)} = 2^m
⇒ (2)^{(3n + 10)} = 2^m
if p^X = p^Y then X will be equal to Y. means X = Y;
On comparing, we get
3n + 10 = m
⇒ m = 3n + 10;