We need to simplify the expression for the square root of \(\frac{x^2 - 4}{x^4}\).

### Step-by-Step Solution:

The expression is:
\[
\sqrt{\frac{x^2 - 4}{x^4}}
\]

#### Step 1: Factor the numerator
Notice that \(x^2 - 4\) can be factored using the difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2)
\]

So the expression becomes:
\[
\sqrt{\frac{(x - 2)(x + 2)}{x^4}}
\]

#### Step 2: Apply the square root to both the numerator and denominator
We can apply the square root separately to the numerator and the denominator:
\[
\frac{\sqrt{(x - 2)(x + 2)}}{\sqrt{x^4}}
\]

#### Step 3: Simplify the denominator
The square root of \(x^4\) is:
\[
\sqrt{x^4} = x^2 \quad (\text{assuming} \ x > 0)
\]

#### Step 4: Final expression
So the expression simplifies to:
\[
\frac{\sqrt{(x - 2)(x + 2)}}{x^2}
\]

This is the simplified form of the given expression.

Would you like more details or have any questions on this?

Here are some related questions for you to explore:

1. How would the solution change if \(x\) can take negative values?
2. What happens if \(x = 2\) or \(x = -2\)?
3. How would you find the derivative of this simplified expression?
4. What is the domain of the simplified expression?
5. How can this expression be rewritten if \(x^2 - 4\) were not factorable?

**Tip:** When simplifying expressions with square roots, always check for factorizations such as difference of squares to make the process easier.

Simplify the expression \(\sqrt{\frac{x^2 - 4}{x^4}}\) with this detailed step-by-step solution. Learn key algebraic concepts such as factoring, difference of squares, and radical expressions. Suitable for students in Grades 9-11.

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