1. **State the problem:** Simplify the expression $$(x^3 y^{-3}) (2 x^5 y^4)^3$$. 2. **Recall the rules:** - When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{mn}$$. - When multiplying like bases, add the exponents: $$a^m \cdot a^n = a^{m+n}$$. - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^m}$$. 3. **Apply the power to the second term:** $$(2 x^5 y^4)^3 = 2^3 x^{5 \times 3} y^{4 \times 3} = 8 x^{15} y^{12}$$. 4. **Rewrite the expression:** $$(x^3 y^{-3}) (8 x^{15} y^{12})$$. 5. **Multiply like bases by adding exponents:** $$x^{3 + 15} y^{-3 + 12} = x^{18} y^{9}$$. 6. **Include the constant factor:** $$8 x^{18} y^{9}$$. 7. **Identify the student's mistake:** The student incorrectly wrote $2^3$ as 6 instead of 8. 8. **Correct answer:** $$8 x^{18} y^{9}$$.