Question: Simplify so that no negative exponents appear in the final result. Assume all variables represent…

Simplify so that no negative exponents appear in the final result. Assume all variables represent nonzero numbers.

\[ \left( \frac{2p}{q^2} \right)^2 \left( \frac{3p^5}{q^{-5}} \right)^{-1} \]

\[ \left( \frac{2p}{q^2} \right)^2 \left( \frac{3p^5}{q^{-5}} \right)^{-1} = \boxed{\phantom{xxxx}} \]

Solution

Simplify the given expression step by step. Start with the expression: \[ \left( \frac{2p}{q^2} \right)^2 \left( \frac{3p^5}{q-5} \right)^{-1} \] Simplify each part separately. Begin with \(\left( \frac{2p}{q^2} \right)^2\): \[ \left( \frac{2p}{q^2} \right)^2 = \frac{(2p)^2}{(q^2)^2} = \frac{4p^2}{q^4} \] Next, simplify \(\left( \frac{3p^5}{q-5} \right)^{-1}\) by taking the reciprocal: \[ \left( \frac{3p^5}{q-5} \right)^{-1} = \frac{q-5}{3p^5} \] Combine the two results: \[ \frac{4p^2}{q^4} \times \frac{q-5}{3p^5} = \frac{4p^2(q-5)}{3p^5q^4} \] Simplify the expression by canceling out common terms: \[ = \frac{4(q-5)}{3p^3q^4} \] The simplified expression is: \[ \frac{4(q-5)}{3p^3q^4} \]