What is the value of a⁵ × a⁷?

Third power of 2 = 2³ = 2 × 2 × 2 = 8
And second power of 3 = 3² = 3 × 3 = 9
? Difference = 9 ? 8 = 1

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.

( 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
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

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;

Third power of 4 = 4³ = 4 × 4 × 4 = 64

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 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 .

^L?3 = M^1/L = Surd of L^th order.