Can u pls transform this [ \sqrt{x^2} = \pm \sqrt{16} ] [ x = \pm 4 ]

The transformation from \(\sqrt{x^2} = \pm \sqrt{16}\) to \(x = \pm 4\) is based on the properties of square roots.

### Step-by-Step Transformation:

1. **Simplify the square roots**:
   - The left side: \(\sqrt{x^2}\) simplifies to \(|x|\), which represents the absolute value of \(x\).
   - The right side: \(\sqrt{16}\) simplifies to \(4\).

   So the equation becomes:
   \[
   |x| = \pm 4
   \]

2. **Interpret the absolute value**:
   - The equation \(|x| = 4\) implies that \(x\) can be either \(4\) or \(-4\).
   - The equation \(|x| = -4\) does not have a real solution because the absolute value of any real number cannot be negative.

   Therefore, we discard \(|x| = -4\), leaving us with:
   \[
   |x| = 4
   \]

3. **Resolve the absolute value**:
   - The equation \(|x| = 4\) translates to two possible solutions: \(x = 4\) or \(x = -4\).

Thus, the equation \(\sqrt{x^2} = \pm \sqrt{16}\) simplifies to:
\[
x = \pm 4
\]

Would you like further details or have any other questions?

Here are 8 related questions for further exploration:

1. What is the general rule for simplifying \(\sqrt{x^2}\)?
2. Why is the equation \(|x| = -4\) not valid in the real number system?
3. How do you solve absolute value equations like \(|x| = a\)?
4. Can you explain how to handle square roots of negative numbers in the complex number system?
5. What is the significance of the \(\pm\) symbol in equations?
6. How would the solution change if the original equation had been \(\sqrt{x^2} = \sqrt{16}\) without the \(\pm\) sign?
7. How do absolute value functions graphically represent their solutions?
8. How do you solve quadratic equations that yield non-real roots?

**Tip:** Always remember that the square root of a square gives the absolute value, not just the original variable.

Explore the transformation from \(\sqrt{x^2} = \pm \sqrt{16}\) to \(x = \pm 4\) with detailed step-by-step explanations. Learn about square roots, absolute value equations, and their significance in algebraic contexts. Suitable for high school students and anyone interested in fundamental algebraic concepts.

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