1. **State the problem:** Identify which expressions are equivalent to $\frac{m^5}{n^5}$ and multiply the binomials $(x - 1.2)(-0.2x)$. 2. **Equivalent expressions to $\frac{m^5}{n^5}$:** Recall the property of exponents: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ and $a^{-m} = \frac{1}{a^m}$. 3. **Check each expression:** - $(mn^{-1})^5 = m^5 (n^{-1})^5 = m^5 n^{-5} = \frac{m^5}{n^5}$ (Equivalent) - $(mn)^5 = m^5 n^5$ (Not equivalent) - $[(mn)^2]^{-3} = (m^2 n^2)^{-3} = m^{-6} n^{-6} = \frac{1}{m^6 n^6}$ (Not equivalent) - $(mn)^{-5} = m^{-5} n^{-5} = \frac{1}{m^5 n^5}$ (Not equivalent) - $m^5 n^{-5} = \frac{m^5}{n^5}$ (Equivalent) 4. **Multiply $(x - 1.2)(-0.2x)$:** Use distributive property: $$ (x)(-0.2x) + (-1.2)(-0.2x) = -0.2x^2 + 0.24x $$ 5. **Final answers:** - Equivalent expressions: $(mn^{-1})^5$, $m^5 n^{-5}$ - Product: $-0.2x^2 + 0.24x$