Given that
(0.00032)^2/5 = (32/100000)^2/5
Solve the equation by algebra Law
= (2⁵/10⁵)^2/5
= {(2/10)⁵}^2/5
= (2/10)^5x2/5
= (1/5)² = 1/25

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

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

?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

? 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

Given, xyz = 1, a^x = b, b^y = c
Now, b = a^x
Apply the law of algebra .
Take the power y of both side equation.
? b^y= a^xy
Take the power z of both side equation.
? b^yz = a^xyz
? ( b^y )^z = a^xyz
Replace the b^y with c and xyz with 1 as per given equation,
? c^z = 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
? (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;

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;

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

( 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