1. The problem asks to simplify the expression $$\frac{32^3}{32^8}$$ and compare it to the options given. 2. Recall the rule for dividing powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. Apply this rule to the expression: $$\frac{32^3}{32^8} = 32^{3-8} = 32^{-5}$$ 4. Now, let's analyze the options: - Option 1: $$32^5$$ - Option 2: $$32^{-5}$$ - Option 3: $$\frac{1}{32^5}$$ - Option 4: $$\frac{1}{32^{-5}}$$ 5. Note that $$32^{-5} = \frac{1}{32^5}$$, so Options 2 and 3 are equivalent. 6. Since the simplified form is $$32^{-5}$$, the correct options are Option 2 and Option 3. 7. To confirm, $$\frac{1}{32^{-5}} = 32^5$$, which is Option 4. Final answer: $$\frac{32^3}{32^8} = 32^{-5} = \frac{1}{32^5}$$, so Options 2 and 3 are correct.