1. **State the problem:** Simplify the expression $$(4^7)^5$$. 2. **Recall the exponent rule:** When raising a power to another power, multiply the exponents: $$ (a^m)^n = a^{m \times n} $$. 3. **Apply the rule:** $$ (4^7)^5 = 4^{7 \times 5} = 4^{35} $$. 4. **Conclusion:** The simplified form is $$4^{35}$$. 1. **State the problem:** Simplify the expression $$4^{-3} \times 3^{-3}$$. 2. **Recall the negative exponent rule:** $$a^{-m} = \frac{1}{a^m}$$. 3. **Rewrite each term:** $$4^{-3} = \frac{1}{4^3}$$ and $$3^{-3} = \frac{1}{3^3}$$. 4. **Multiply the fractions:** $$\frac{1}{4^3} \times \frac{1}{3^3} = \frac{1}{4^3 \times 3^3}$$. 5. **Combine the bases inside the product:** $$4^3 \times 3^3 = (4 \times 3)^3 = 12^3$$. 6. **Final simplified form:** $$\frac{1}{12^3}$$.