**Solution:**

To find the value of $\sqrt{x} - \frac{1}{\sqrt{x}}$, we need to substitute the value of $x$ and simplify the expression. Given that $x = 3\sqrt{8}$, let's proceed with the solution step by step.

**Step 1: Simplify the value of x**

Given $x = 3\sqrt{8}$, we can simplify it as follows:

$x = 3\sqrt{2^3} = 3(2\sqrt{2}) = 6\sqrt{2}$

**Step 2: Substitute the value of x in the expression**

Now, we can substitute the value of $x$ in the expression $\sqrt{x} - \frac{1}{\sqrt{x}}$:

$\sqrt{x} - \frac{1}{\sqrt{x}} = \sqrt{6\sqrt{2}} - \frac{1}{\sqrt{6\sqrt{2}}}$

**Step 3: Simplify the expression**

To simplify the expression, we need to rationalize the denominator. Rationalizing the denominator involves multiplying the numerator and denominator by the conjugate of the denominator.

$\sqrt{6\sqrt{2}} - \frac{1}{\sqrt{6\sqrt{2}}} = \sqrt{6\sqrt{2}} - \frac{1}{\sqrt{6\sqrt{2}}} \times \frac{\sqrt{6\sqrt{2}}}{\sqrt{6\sqrt{2}}}$

Simplifying the denominator:

$\sqrt{6\sqrt{2}} - \frac{1}{\sqrt{6\sqrt{2}}} \times \frac{\sqrt{6\sqrt{2}}}{\sqrt{6\sqrt{2}}} = \sqrt{6\sqrt{2}} - \frac{\sqrt{6\sqrt{2}}}{6\sqrt{2}}$

Simplifying the expression:

$\sqrt{6\sqrt{2}} - \frac{\sqrt{6\sqrt{2}}}{6\sqrt{2}} = \sqrt{6\sqrt{2}} - \frac{1}{6}$

**Step 4: Reorder the terms**

To make the expression simpler, we can reorder the terms:

$\sqrt{6\sqrt{2}} - \frac{1}{6} = -\frac{1}{6} + \sqrt{6\sqrt{2}}$

Therefore, the value of $\sqrt{x} - \frac{1}{\sqrt{x}}$ when $x = 3\sqrt{8}$ is $-\frac{1}{6} + \sqrt{6\sqrt{2}}$.