Question: Which is an equivalent expression using a positive exponent?
$$\left( \frac{3}{5} \right)^{-8} \div \left( \frac{3}{5} \right)^{-4}$$
$$\left( \frac{3}{5} \right)^4$$
$$\left( \frac{3}{5} \right)^{12}$$
$$\left( \frac{5}{3} \right)^{32}$$
$$\left( \frac{5}{3} \right)^4$$
1. **State the problem:**
We want to simplify the expression $$\left( \frac{3}{5} \right)^{-8} \div \left( \frac{3}{5} \right)^{-4}$$ and express it using a positive exponent.
2. **Recall the exponent division rule:**
When dividing powers with the same base, subtract the exponents:
$$a^m \div a^n = a^{m-n}$$
3. **Apply the rule:**
$$\left( \frac{3}{5} \right)^{-8} \div \left( \frac{3}{5} \right)^{-4} = \left( \frac{3}{5} \right)^{-8 - (-4)} = \left( \frac{3}{5} \right)^{-8 + 4} = \left( \frac{3}{5} \right)^{-4}$$
4. **Rewrite with a positive exponent:**
Recall that $$a^{-m} = \frac{1}{a^m}$$, so
$$\left( \frac{3}{5} \right)^{-4} = \left( \frac{5}{3} \right)^4$$
5. **Conclusion:**
The equivalent expression using a positive exponent is $$\left( \frac{5}{3} \right)^4$$.
**Answer:** $$\left( \frac{5}{3} \right)^4$$