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.