Questions: Simplify. (u^-4 z^5)(2 x u^2 / z^-1)^-2 Write your answer using only positive exponents.

Solution Steps

To simplify the given expression and write the answer using only positive exponents, follow these steps:

1. Simplify the inner expression \(\left(\frac{2 x u^{2}}{z^{-1}}\right)^{-2}\) by first dealing with the negative exponent.
2. Apply the negative exponent to each term inside the parentheses.
3. Combine the simplified inner expression with the outer expression \(\left(u^{-4} z^{5}\right)\).
4. Simplify the resulting expression by combining like terms and ensuring all exponents are positive.

First, simplify the inner expression \(\left(\frac{2 x u^{2}}{z^{-1}}\right)^{-2}\): \[ \left(\frac{2 x u^{2}}{z^{-1}}\right)^{-2} = \left(2 x u^{2} \cdot z\right)^{-2} = \left(2 x u^{2} z\right)^{-2} \]

Apply the negative exponent to each term inside the parentheses: \[ \left(2 x u^{2} z\right)^{-2} = \frac{1}{(2 x u^{2} z)^{2}} = \frac{1}{4 x^{2} u^{4} z^{2}} \]

Combine the simplified inner expression with the outer expression \(\left(u^{-4} z^{5}\right)\): \[ \left(u^{-4} z^{5}\right) \cdot \frac{1}{4 x^{2} u^{4} z^{2}} = \frac{u^{-4} z^{5}}{4 x^{2} u^{4} z^{2}} \]

Simplify the resulting expression by combining like terms and ensuring all exponents are positive: \[ \frac{u^{-4} z^{5}}{4 x^{2} u^{4} z^{2}} = \frac{z^{5-2}}{4 x^{2} u^{4+4}} = \frac{z^{3}}{4 x^{2} u^{8}} \]

Final Answer