1. **Stating the problem:** We need to simplify the expressions using the rules of exponents. 2. **Recall the exponent rules:** - $a^m \cdot a^n = a^{m+n}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $(a^m)^n = a^{m \cdot n}$ - $a^{-m} = \frac{1}{a^m}$ --- ### a) Simplify $6^4 \cdot 6 \cdot 6^{-4}$ 3. Combine the powers of 6 by adding exponents: $$6^4 \cdot 6^1 \cdot 6^{-4} = 6^{4+1-4}$$ 4. Simplify the exponent: $$6^{\cancel{4}+1-\cancel{4}} = 6^1 = 6$$ --- ### b) Simplify $\frac{5^6 \cdot 5^{-3}}{5^2 \cdot 5}$ 5. Simplify numerator by adding exponents: $$5^{6 + (-3)} = 5^3$$ 6. Simplify denominator by adding exponents: $$5^{2 + 1} = 5^3$$ 7. Write the fraction: $$\frac{5^3}{5^3} = 5^{3-3} = 5^0$$ 8. Recall $5^0 = 1$ --- ### c) Simplify $\frac{(3b)^2 \cdot b^3 \cdot a^{-4}}{b^5 \cdot a^{-5}}$ 9. Expand $(3b)^2$: $$(3b)^2 = 3^2 \cdot b^2 = 9b^2$$ 10. Substitute back: $$\frac{9b^2 \cdot b^3 \cdot a^{-4}}{b^5 \cdot a^{-5}} = \frac{9b^{2+3} a^{-4}}{b^5 a^{-5}} = \frac{9b^5 a^{-4}}{b^5 a^{-5}}$$ 11. Simplify $b$ terms: $$\frac{9 \cancel{b^5} a^{-4}}{\cancel{b^5} a^{-5}} = 9 \cdot \frac{a^{-4}}{a^{-5}} = 9 a^{-4 - (-5)} = 9 a^{1} = 9a$$ --- **Final answers:** - a) $6$ - b) $1$ - c) $9a$