Explanation:
so, we have
(343/1024×32)^(1/5) × 256^(1/2)
we know that we should process first brackets and their content, then exponents, and then multiplications and divisions.
we also know that an exponent has to be applied to all parts and all factors of its base term.
and we know that a fraction "m/n" in an exponent means "to the power of m" and pulling the nth root.
given the nature of this question (homework or test) it is likely that the planned result is a whole or at least a nice short number.
and so I think you made a typo, and the first number should be 243 (and not 343). am I right ?
because 243 = 3⁵, but 343 is "only" 7³ (but not a 5th power of a "nice" number).
so, for
(343/1024×32) we should pull the 5th root.
as mentioned,
343 is not the 5th power of any nice number. but 243 is : 3⁵
1024 = 4⁵
32 = 2⁵
so, the 5th root of this term is
343^(1/5) / 4 × 2 = 343^(1/5) / 2
and for 256 we should pull the square root (the second root).
256 = 2⁸ = 16²
so, the square root is 16.
and the total result is then
343^(1/5) / 2 × 16 = 343^(1/5) × 8 = 25.7127668...
now, if we assume that you actuality meant 243 as first number, we would then have the 5th root of 243 = 3 and get
3 × 8 = 24
as total result.
update : you mentioned that the expected result is 6.
then you made the mentioned typo AND left out some more brackets.
I assume the real expression is then
(243÷(1024×32))^(1/5) × 256^(1/2) =
= (3⁵ ÷ (4⁵ × 2⁵))^(1/5) × (16²)^(1/2) =
= (3 ÷ (4 × 2)) × 16 = 3 ÷ 8 × 16 = 3 × 2 = 6
if your teacher actually left out these brackets, then he/she made a mistake.
then my original solution above (24) is correct. because without the brackets I must not apply the multiplication by 32 as factor of 1024, but must apply it as factor of (243/1024).