1. **State the problem:** Simplify the expression $$\frac{4t^3}{(3t^3)^5}$$ using only positive exponents. 2. **Recall the exponent rules:** - Power of a product: $$(ab)^n = a^n b^n$$ - Power of a power: $$(a^m)^n = a^{mn}$$ - Division of like bases: $$\frac{a^m}{a^n} = a^{m-n}$$ 3. **Apply the power of a product rule:** $$(3t^3)^5 = 3^5 (t^3)^5 = 3^5 t^{3 \times 5} = 3^5 t^{15}$$ 4. **Rewrite the original expression:** $$\frac{4t^3}{3^5 t^{15}}$$ 5. **Simplify the fraction:** $$= \frac{4}{3^5} \times \frac{t^3}{t^{15}}$$ 6. **Apply the division of like bases rule:** $$\frac{t^3}{t^{15}} = t^{3-15} = t^{-12}$$ 7. **Rewrite with positive exponents:** $$t^{-12} = \frac{1}{t^{12}}$$ 8. **Combine all parts:** $$\frac{4}{3^5} \times \frac{1}{t^{12}} = \frac{4}{3^5 t^{12}}$$ 9. **Calculate $3^5$:** $$3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243$$ 10. **Final simplified expression:** $$\boxed{\frac{4}{243 t^{12}}}$$