Determine the value of y*y*, if x*x* is minus, 10−10.

y, equals, open vertical bar, x, minus, 5, close vertical bar

*y*=

∣*x*−5∣

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 \( y \):

   \[
   y = \sqrt{(-5)^2 + \left(\sqrt{10^{-10}}\right)^2}
   = \sqrt{25 + 10^{-10}}
   \]
   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.

Determine the value of y*y, if x*x is -10^-10. y = |x - 5|

Solve a complex number problem where x*x = -10^-10 and y = |x - 5|. Explore key concepts like the magnitude of complex numbers and absolute value. Suitable for advanced math students in Grades 11-12 or higher.

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