1. Simplify $3^4 \times 3^2$. Using the product of powers rule: $a^m \times a^n = a^{m+n}$, we get $3^{4+2} = 3^6$. Calculating $3^6 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729$. 2. Simplify $5^6 \div 5^3$. Using the quotient of powers rule: $a^m \div a^n = a^{m-n}$, we get $5^{6-3} = 5^3$. Calculating $5^3 = 5 \times 5 \times 5 = 125$. 3. Simplify $(2^3)^2$. Using the power of a power rule: $(a^m)^n = a^{m \times n}$, we get $2^{3 \times 2} = 2^6$. Calculating $2^6 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64$. 4. Simplify $7^0$. Any nonzero number to the zero power is 1, so $7^0 = 1$. 5. Simplify $9^1$. Any number raised to the power 1 is itself, so $9^1 = 9$. 6. Simplify $1 \div a^3$. This can be rewritten as $a^{-3}$ (negative exponent rule). 7. $a^{-5}$ is already simplified and equals $\frac{1}{a^5}$. 8. Simplify $\frac{1}{3^2} \times 3^2$. Rewrite as $3^{-2} \times 3^2$. Using product of powers: $3^{-2+2} = 3^0 = 1$. Final answers: 1. $729$ 2. $125$ 3. $64$ 4. $1$ 5. $9$ 6. $a^{-3}$ 7. $a^{-5}$ 8. $1$