Explanation:
remember the rules about exponents :
a^x × a^y = a^(x + y)
a^x / a^y = a^(x - y)
(a^x)^y = a^(x×y)
a^(-x) = 1/(a^x)
a^(1/x) = xth root of a
an overall exponent always applies to all factors.
so, we have
((3m^(-1) × n²)⁴) / ((2m^(-2) × n)³)
let's start with the top alone first :
((3m^(-1) × n²)⁴) = 3⁴ × m^(-1×4) × n^(2×4) =
= 81 × m^(-4) × n⁸
and now the bottom part :
((2m^(-2) × n)³) = 2³ × m^(-2×3) × n^(1×3) =
= 8 × m^(-6) × n³
now, when we bring them together again as numerator and denominator we see
(81 × m^(-4) × n⁸) / (8 × m^(-6) × n³)
we can do the simplification factor type by factor type :
81/8 is clear, right ?
m^(-4)/m^(-6) = m^(‐4 ‐ -6) = m^(‐4 + 6) = m²
n⁸/n³ = n^(8 - 3) = n⁵
so, the result is
(81/8) × m²n⁵ = 81m²n⁵/8