1. **Problem statement:** Simplify each expression, leaving the answer in index form. 2. **Important rules:** - When multiplying powers with the same base, add the exponents: $a^m \times a^n = a^{m+n}$. - When dividing powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$. - When raising a power to another power, multiply the exponents: $(a^m)^n = a^{mn}$. 3. **Solutions:** **a.** $4 \times 4^x$ Rewrite $4$ as $4^1$: $$4^1 \times 4^x = 4^{1+x}$$ **b.** $\frac{3^y}{9}$ Rewrite $9$ as $3^2$: $$\frac{3^y}{3^2} = 3^{y-2}$$ **c.** $(7^{2z})^6$ Multiply exponents: $$7^{2z \times 6} = 7^{12z}$$ **d.** $\frac{5^{2x} \times 5}{5^x}$ Rewrite $5$ as $5^1$ and multiply numerator: $$\frac{5^{2x} \times 5^1}{5^x} = \frac{5^{2x+1}}{5^x} = 5^{(2x+1)-x} = 5^{x+1}$$ **e.** $\frac{(3^a)^4}{3^b}$ Simplify numerator: $$(3^a)^4 = 3^{4a}$$ Divide powers: $$\frac{3^{4a}}{3^b} = 3^{4a - b}$$ **Final answers:** - a) $4^{1+x}$ - b) $3^{y-2}$ - c) $7^{12z}$ - d) $5^{x+1}$ - e) $3^{4a - b}$