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:** $$2^2 \times 2^{\frac{1}{2}} = 2^{2 + \frac{1}{2}} = 2^{\frac{4}{2} + \frac{1}{2}} = 2^{\frac{5}{2}}$$. 4. **Rewrite the expression:** $$\frac{2^{\frac{5}{2}}}{2^{-3}}$$. 5. **Apply division rule:** $$2^{\frac{5}{2} - (-3)} = 2^{\frac{5}{2} + 3} = 2^{\frac{5}{2} + \frac{6}{2}} = 2^{\frac{11}{2}}$$. 6. **Final simplified form:** $$2^{\frac{11}{2}}$$ or equivalently $$\sqrt{2^{11}} = \sqrt{2048}$$. Thus, the simplified expression is $$2^{\frac{11}{2}}$$.