1. **State the problem:** Calculate the value of $\sqrt{2}^5$. 2. **Recall the properties of exponents:** For any positive number $a$ and rational exponents $m$ and $n$, we have $\left(a^{m}\right)^n = a^{m \times n}$. 3. **Rewrite the expression:** Since $\sqrt{2} = 2^{\frac{1}{2}}$, then $$\sqrt{2}^5 = \left(2^{\frac{1}{2}}\right)^5$$ 4. **Apply the exponent rule:** $$\left(2^{\frac{1}{2}}\right)^5 = 2^{\frac{1}{2} \times 5} = 2^{\frac{5}{2}}$$ 5. **Simplify the exponent:** $$2^{\frac{5}{2}} = 2^{2 + \frac{1}{2}} = 2^2 \times 2^{\frac{1}{2}}$$ 6. **Evaluate each term:** $$2^2 = 4$$ $$2^{\frac{1}{2}} = \sqrt{2}$$ 7. **Combine the results:** $$4 \times \sqrt{2}$$ **Final answer:** $$\sqrt{2}^5 = 4 \sqrt{2}$$