Final answer:
The simplified form of the given mathematical expression, applying properties of exponents, is y^(-1/8).
Step-by-step explanation:
To simplify the given expression, we can use the properties of exponents, which tell us that (a^(m))^n = a^(mn), a^m / a^n = a^(m-n), and a^-n = 1/a^n. Thus, we first simplify the numerator to have (z^(4*1/8)y^(1/8)) = z^(1/2)y^(1/8) and we simplify the denominator to have z^(1/2)y^(1/4). Then we divide, using the properties of exponents to subtract the powers of like variables, resulting in: (z^(1/2)y^(1/8)) / (z^(1/2)y^(1/4)) = z^(1/2 - 1/2)y^(1/8 - 1/4) = y^0 * y^(-1/8) = y^(-1/8). Therefore, the simplified form of the expression is y^(-1/8).
Learn more about Properties of Exponents