1. **State the problem:** Determine which statement is true among the following comparisons involving exponents: $$\frac{3^{-15}}{3^{7}} \quad \text{and} \quad (3^{-8}) \cdot (3^{-9})$$ Options: - A: $$\frac{3^{-15}}{3^{7}} < (3^{-8}) \cdot (3^{-9})$$ - B: $$\frac{3^{-15}}{3^{7}} > (3^{-8}) \cdot (3^{-9})$$ - C: $$\frac{3^{-15}}{3^{7}} = (3^{-8}) \cdot (3^{-9})$$ 2. **Recall exponent rules:** - Division of same bases: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$ - Multiplication of same bases: $$a^{m} \cdot a^{n} = a^{m+n}$$ 3. **Simplify the left side:** $$\frac{3^{-15}}{3^{7}} = 3^{-15 - 7} = 3^{-22}$$ 4. **Simplify the right side:** $$(3^{-8}) \cdot (3^{-9}) = 3^{-8 + (-9)} = 3^{-17}$$ 5. **Compare the two expressions:** We have $$3^{-22}$$ and $$3^{-17}$$. Since the base 3 is greater than 1, the function $$3^{x}$$ is increasing, so the smaller exponent corresponds to a smaller value. Because $$-22 < -17$$, it follows that: $$3^{-22} < 3^{-17}$$ 6. **Conclusion:** $$\frac{3^{-15}}{3^{7}} = 3^{-22} < 3^{-17} = (3^{-8}) \cdot (3^{-9})$$ Therefore, **Option A is true.**