(x^{2}z)^{1/4}/x^{-1/2}z^{1/2}
First we want to simplify the expression in the numerator. Since the term (x^{2}z) is being raised to the 1/4 power, we multiply the exponents of each variable to give us a simplified power. The denominator is already simplified. This gives us
x^{2*1/4}z^{1/4}/x^{-1/2}z^{1/2}
x^{1/2}z^{1/4}/x^{-1/2}z^{1/2}
To further simplify from this point, we make use of the fact that x^{A}/x^{B}=x^{A-B}. That is, the power of exponential terms divided by terms with the same base is equal to the difference in the exponents. We do this for both
x and z.
x^{1/2-(-1/2)}z^{1/4-1/2}
We recognize that the subtraction of a negative term in a power of x leads us to add the two terms, and we find a common denominator, 4, to subtract the powers of z. We find
x^{1}z^{-1/4}
The power of 1 in the x term gives us our base, x, and the power of -1/4 in the z term can be rewritten as a positive exponent in the denominator of our simplified expression. This gives the answer
x/z^{1/4}