Step-by-step explanation:
you know that a negative exponent means 1/...
and you know that with a fraction as exponent the numerator (top part) is an actual exponent (to the power of), and the denominator (bottom part) of actually the root to that degree.
e.g. the exponent of 1/2 means the square root.
also : (a^b)^c = a^(bc)
and remember basic fraction arithmetic :
a/b / c/d = ad / bc = a/b × d/c
so, now there should not be any mysteries left here (and I am adding some missing brackets - hopefully you missed nothing else) :
((81/16)^(3/4) × (25/9)^(-3/4)) / ((5/2)^(-3))
first we react to the minus signs in the exponents. so this is the same as :
((81/16)^(3/4) × (9/25)^(3/4)) / ((2/5)³)
this is the same as
(((81/16) × (9/25))^(3/4)) × (5/2)³
now we recognize that all the basic numbers on the left are squared numbers (power of 2).
so, this is the same as
(((9/4)² × (3/5)²)^(3/4)) × (5/2)³ =
= ((((9/4) × (3/5))²)^(3/4)) × (5/2)³ =
= (((9/4) × (3/5))^(2×(3/4))) × (5/2)³ =
= (((9/4) × (3/5))^(2×3/4)) × (5/2)³ =
= (((9/4) × (3/5))^(3/2)) × (5/2)³ =
= ((((9/4) × (3/5))^(1/2)) × (5/2))³ =
= (((3/2) × sqrt(3/5)) × (5/2))³ =
= ((3/2) × (5/2) × sqrt(3/5))³ =
= ((15/4) × sqrt(3/5))³
(sqrt(3/5))³ = sqrt(3/5)×sqrt(3/5)×sqrt(3/5) =
= (3/5) × sqrt(3/5)
our expression is then therefore equal to
(15/4)³ × (sqrt(3/5))³ =
(15/4)³ × (3/5) × sqrt(3/5) =
(15/4)×(15/4)×(15/4) × (3/5) × sqrt(3/5) =
(3/4)×(15/4)×(15/4) × 3 × sqrt(3/5) =
(9/4)×(15/4)×(15/4) × sqrt(3/5) = (2025/64) × sqrt(3/5) =
(45/8)² × sqrt(3/5)
not a super-nice result.
so, I suspect there was another mistake in the original expression here.
