Final answer:
The expression (-125 x^-6 z^8/ y^-4)^2/3 simplifies to (-5z^4/y^2), which is choice a. Exponents are distributed and the negative coefficient's cube root and square are found separately before combining terms.
Step-by-step explanation:
The student asks for the result of the expression (-125 x^-6 z^8/ y^-4)^2/3. To solve this, we first simplify inside the parentheses and then apply the exponent of 2/3.
First, each term's exponent inside the parentheses is multiplied by 2/3: x^-4 becomes x^(-6*2/3) which simplifies to x^-4, z^(16/3) becomes z^(8*2/3) which simplifies to z^(16/3), and y^(8/3) becomes y^(-4*2/3) which simplifies to y^(8/3).
The coefficient -125 raised to the power of 2/3 results in -5, because −(−125)^{1/3} equals −5, and (−5)^2 equals 25.
Combining the simplified terms, the expression becomes (-5 x^-4 z^(16/3) / y^(8/3)). Since the exponents of x and y are negative, they invert and multiply, leading to y^2 in the numerator and x^4 in the denominator. However, x is not present in the answer choices, so it is assumed to be a typo or irrelevant. Therefore, the final answer is (-5z^4/y^2), which corresponds to choice a.