1. **State the problem:** Solve the exponential equation $$2^{3x - 2} = 8^{\frac{1}{2x}}.$$\n\n2. **Rewrite the bases:** Note that 8 can be written as a power of 2: $$8 = 2^3.$$ So the equation becomes $$2^{3x - 2} = (2^3)^{\frac{1}{2x}}.$$\n\n3. **Apply the power of a power rule:** $$ (a^m)^n = a^{mn} $$, so we get $$2^{3x - 2} = 2^{3 \cdot \frac{1}{2x}} = 2^{\frac{3}{2x}}.$$\n\n4. **Set the exponents equal:** Since the bases are the same and nonzero, the exponents must be equal: $$3x - 2 = \frac{3}{2x}.$$\n\n5. **Solve the equation:** Multiply both sides by $$2x$$ to clear the denominator: $$2x(3x - 2) = 3.$$\n\n6. **Expand:** $$6x^2 - 4x = 3.$$\n\n7. **Bring all terms to one side:** $$6x^2 - 4x - 3 = 0.$$\n\n8. **Use the quadratic formula:** For $$ax^2 + bx + c = 0$$, $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $$a=6$$, $$b=-4$$, $$c=-3$$.\n\n9. **Calculate the discriminant:** $$\Delta = (-4)^2 - 4 \cdot 6 \cdot (-3) = 16 + 72 = 88.$$\n\n10. **Find the roots:** $$x = \frac{4 \pm \sqrt{88}}{12} = \frac{4 \pm 2\sqrt{22}}{12} = \frac{2 \pm \sqrt{22}}{6}.$$\n\n**Final answer:** $$x = \frac{2 + \sqrt{22}}{6}$$ or $$x = \frac{2 - \sqrt{22}}{6}.$$