Exponents and Polynomials
Power, product, and quotient rules with negative exponents

Simplify.
\[
\left(u z^{5}\right)\left(\frac{x^{3} u^{-1}}{2 z^{3}}\right)^{-3}
\]

Write your answer using only positive exponents.

Step 1 ：The problem is asking to simplify the given expression and rewrite it using only positive exponents.

Step 2 ：The first step is to simplify the expression inside the parentheses. The negative exponent means that we need to take the reciprocal of the base.

Step 3 ：Then, we can apply the power rule, which states that when raising a quotient to a power, we raise both the numerator and the denominator to that power.

Step 4 ：After that, we can simplify the expression by canceling out common terms in the numerator and the denominator.

Step 5 ：Finally, we can apply the product rule, which states that when multiplying terms with the same base, we add the exponents.

Step 6 ：The expression has been simplified to \(8u^{4}z^{14}/x^{9}\). This expression is already written using only positive exponents. Therefore, this is the final answer.

Step 7 ：Final Answer: \(\boxed{8u^{4}z^{14}/x^{9}}\)