Question: What is the value of this expression?
$$\frac{3^2 \times 3^{-5}}{\frac{1}{243}} \quad \frac{1}{27} \quad \frac{1}{9} \quad 27$$
1. **State the problem:** We need to find the value of the expression $$\frac{3^2 \times 3^{-5}}{\frac{1}{243}}$$.
2. **Recall the laws of exponents:** When multiplying powers with the same base, add the exponents: $$a^m \times a^n = a^{m+n}$$.
3. **Simplify the numerator:**
$$3^2 \times 3^{-5} = 3^{2 + (-5)} = 3^{-3}$$
4. **Rewrite the denominator:**
Since $$243 = 3^5$$, then
$$\frac{1}{243} = 3^{-5}$$
5. **Rewrite the entire expression:**
$$\frac{3^{-3}}{3^{-5}}$$
6. **Divide powers with the same base:**
$$\frac{a^m}{a^n} = a^{m-n}$$, so
$$3^{-3 - (-5)} = 3^{-3 + 5} = 3^2$$
7. **Calculate the final value:**
$$3^2 = 9$$
**Final answer:** $$9$$