The problem provided is:

$\frac{1}{2} \cdot \sqrt[3]{2^{x+1}} = \frac{1}{\sqrt{8}}$

We need to find the value of $x$ that satisfies this equation. Let's solve it step by step.

Step 1: Simplify the equation

First, let's express $\sqrt{8}$ in a simpler form. We know:

$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$

Thus, the right-hand side becomes:

$\frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}}$

So the equation becomes:

$\frac{1}{2} \cdot \sqrt[3]{2^{x+1}} = \frac{1}{2\sqrt{2}}$

Step 2: Eliminate the common factor

Both sides of the equation have a factor of $\frac{1}{2}$, so we can cancel this out:

$\sqrt[3]{2^{x+1}} = \frac{1}{\sqrt{2}}$

Step 3: Simplify the right-hand side

Notice that $\sqrt{2}$ can be rewritten as $2^{\frac{1}{2}}$. Therefore:

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

So the equation becomes:

$\sqrt[3]{2^{x+1}} = 2^{-\frac{1}{2}}$

Step 4: Solve for $x$

Since both sides of the equation are powers of 2, we can set the exponents equal to each other. The equation simplifies to:

$\frac{x+1}{3} = -\frac{1}{2}$

Step 5: Solve for $x$

To solve for $x$, multiply both sides by 3:

$x+1 = -\frac{3}{2}$

Now, subtract 1 from both sides:

$x = -\frac{3}{2} - 1 = -\frac{3}{2} - \frac{2}{2} = -\frac{5}{2}$

Thus, the value of $x$ is:

$x = -\frac{5}{2}$

Would you like further details or have any questions?

Here are 5 related questions that might help deepen your understanding:

1. How can you verify that the solution $x = -\frac{5}{2}$ is correct by substituting it back into the original equation?
2. What are the general properties of logarithms that could be applied to solve similar exponential equations?
3. How does changing the base of the exponent affect the solution of the equation?
4. How would the solution change if the equation were $\sqrt[3]{2^{x+2}} = \frac{1}{\sqrt{2}}$?
5. How can the equation be solved using logarithms instead of exponent rules?

Tip: When solving equations involving roots and exponents, it's often useful to express everything in terms of the same base to simplify the process.