Simplify the expression. Write the answer with positive exponents. Assume all variables represent positive numbers.

(x

^{2}z)^{1/4}/x^{-1/2}z^{1/2}Simplify the expression. Write the answer with positive exponents. Assume all variables represent positive numbers.

(x^{2}z)^{1/4}/x^{-1/2}z^{1/2}

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(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}

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