Explanation:
remember :
an exponent on an exponent means the multiplication of both exponents.
an exponent on top of a multiplication or division applies to all terms and factors in that multiplication or division.
an exponent of n (or n/1) means "to the power of n".
an exponent of 1/n means the nth root.
a negative exponent means 1/...
((8a³) / (27b⁶))^(‐4/3) is therefore the 3rd (cubic) root of the upside-down division to the power of 4 :
1 / (((8a³) / (27b⁶))^(4/3)) =
= (((27b⁶)^(4/3)) / ((8a³)^(4/3))) =
= ((27^(4/3) × b^(6×4/3)) / (8^(4/3) × a^(3×4/3))) =
= (((3³)^(4/3) × b^(6×4/3)) / ((2³)^(4/3) × a^(3×4/3))) =
= ((3^(3×4/3) × b^(6×4/3)) / (2^(3×4/3) × a^(3×4/3))) =
= ((3^(4) × b^(2×4)) / (2^(4) × a^(4))) =
= ((3^(4) × b^(8)) / (2^(4) × a^(4))) =
= ((3⁴b⁸) / (2⁴a⁴)) =
= (81b⁸) / (16a⁴) =
= ((3b²) / (2a))⁴