Let's break down the given expression step by step to determine the value of $y \cdot y$, where $y = |x - 5|$, and it is also given that $x \cdot x = -10^{-10}$.

1. The given expression $x \cdot x = -10^{-10}$ cannot be real, because the square of any real number $x$ is non-negative. So, this suggests that $x$ might be a complex number, since only a complex number can have a negative square. Let’s denote $x$ as a complex number $x = i \cdot \sqrt{10^{-10}}$, where $i$ is the imaginary unit.

2. Now we need to find $y = |x - 5|$. If $x = i \cdot \sqrt{10^{-10}}$, then we substitute this into the expression: $y = |i \cdot \sqrt{10^{-10}} - 5|$ The magnitude (absolute value) of a complex number $a + bi$ is $\sqrt{a^2 + b^2}$. Here, $a = -5$ and $b = \sqrt{10^{-10}}$, so we can calculate $y$:

   = \sqrt{25 + 10^{-10}}$$ Since $$10^{-10}$$ is very small compared to 25, we can approximate: $$y \approx \sqrt{25} = 5$$

3. Finally, we need to compute $y \cdot y$: $y \cdot y = 5 \cdot 5 = 25$

Thus, the value of $y \cdot y$ is approximately 25.

Would you like any more details or further clarifications?

Here are 5 follow-up questions:

1. What is the magnitude of a complex number?
2. Why is the square of a real number always non-negative?
3. Can you explain how to simplify square roots with very small numbers?
4. How does the imaginary unit $i$ affect the calculations in this problem?
5. What are some properties of absolute values when applied to complex numbers?

Tip: When dealing with small approximations like $10^{-10}$, consider the impact on larger numbers in the calculation for accuracy.