1. **State the problem:** Simplify the expression $$\frac{2^{-2} \times 2^{\frac{1}{2}}}{2^{-3}}$$. 2. **Recall the exponent 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: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Apply multiplication rule to the numerator:** $$2^{-2} \times 2^{\frac{1}{2}} = 2^{-2 + \frac{1}{2}} = 2^{-\frac{4}{2} + \frac{1}{2}} = 2^{-\frac{3}{2}}$$. 4. **Rewrite the expression:** $$\frac{2^{-\frac{3}{2}}}{2^{-3}}$$. 5. **Apply division rule:** $$2^{-\frac{3}{2} - (-3)} = 2^{-\frac{3}{2} + 3} = 2^{-\frac{3}{2} + \frac{6}{2}} = 2^{\frac{3}{2}}$$. 6. **Final simplified form:** $$2^{\frac{3}{2}}$$. This means the expression simplifies to $$2^{\frac{3}{2}}$$, which can also be written as $$\sqrt{2^3} = \sqrt{8} = 2\sqrt{2}$$ if a radical form is preferred.