1. **Problem:** Given that $10^x = 64$, find the value of $10^{\frac{x}{2} + 1}$.\n\n2. **Formula and rules:** We use the properties of exponents: $10^{a+b} = 10^a \times 10^b$ and $10^{ka} = (10^a)^k$.\n\n3. **Step 1:** Express $10^{\frac{x}{2} + 1}$ as $10^{\frac{x}{2}} \times 10^1$.\n\n4. **Step 2:** Since $10^x = 64$, then $10^{\frac{x}{2}} = \sqrt{10^x} = \sqrt{64} = 8$.\n\n5. **Step 3:** Therefore, $10^{\frac{x}{2} + 1} = 8 \times 10 = 80$.\n\n**Final answer:** $80$