To find the minimum value of $x^2 + y^2 + z^2$ given that $xyz = 8$, we can apply some techniques from optimization, such as the AM-GM inequality or Lagrange multipliers. Here, we'll proceed using AM-GM inequality for simplicity.

Step 1: Use Symmetry and AM-GM Inequality

We want to minimize $x^2 + y^2 + z^2$ under the constraint $xyz = 8$. By symmetry, let's assume that $x = y = z$. This assumption simplifies the problem and is often a good starting point when variables appear symmetrically.

Step 2: Simplify the Condition $xyz = 8$

If $x = y = z$, then the condition $xyz = 8$ becomes: $x^3 = 8$ Solving for $x$, we get: $x = 2$

Since $x = y = z$, we have: $y = 2, \quad z = 2$

Step 3: Calculate $x^2 + y^2 + z^2$

Now, substitute $x = 2$, $y = 2$, and $z = 2$ into the expression $x^2 + y^2 + z^2$: $x^2 + y^2 + z^2 = 2^2 + 2^2 + 2^2 = 4 + 4 + 4 = 12$

Conclusion:

The minimum value of $x^2 + y^2 + z^2$ is 12, and this occurs when $x = y = z = 2$.

Would you like more details on why symmetry works here or any other explanation?

---

Here are 5 related questions for further practice:

1. What is the minimum value of $x^2 + y^2$ given that $xy = 4$?
2. Solve for the minimum of $x^2 + y^2 + z^2$ subject to $xyz = -8$.
3. How can the method of Lagrange multipliers be applied to this problem?
4. Find the maximum value of $x + y + z$ given that $xyz = 8$.
5. Can we find the same minimum value using derivatives instead of AM-GM?

Tip: The AM-GM inequality is useful for finding minimum or maximum values when you have symmetric expressions involving multiplication and addition.