We are given the following expression:

$\frac{\sqrt{1 - x^x} + \frac{x^2}{\sqrt{1 - x^2}}}{1 - x^2}$

Let's break this down step-by-step:

Step 1: Simplify the numerator

The numerator of the expression is:

$\sqrt{1 - x^x} + \frac{x^2}{\sqrt{1 - x^2}}$

Step 2: Combine terms over a common denominator

To simplify the entire expression, we'll first rewrite it as:

$\frac{\sqrt{1 - x^x} + \frac{x^2}{\sqrt{1 - x^2}}}{1 - x^2}$

Now, we combine the two terms in the numerator over a common denominator. The term $\frac{x^2}{\sqrt{1 - x^2}}$ has a denominator of $\sqrt{1 - x^2}$, so we’ll need to express both terms with a common denominator.

Let’s combine the terms in the numerator first. To add them, we write the first term, $\sqrt{1 - x^x}$, as:

$\sqrt{1 - x^x} = \frac{\sqrt{1 - x^x} \cdot \sqrt{1 - x^2}}{\sqrt{1 - x^2}}$

Now the numerator looks like:

$\frac{\sqrt{1 - x^x} \cdot \sqrt{1 - x^2}}{\sqrt{1 - x^2}} + \frac{x^2}{\sqrt{1 - x^2}}$

Combining these two terms gives:

$\frac{\sqrt{1 - x^x} \cdot \sqrt{1 - x^2} + x^2}{\sqrt{1 - x^2}}$

Step 3: Place everything over the denominator $1 - x^2$

Now, the whole expression looks like:

$\frac{\frac{\sqrt{1 - x^x} \cdot \sqrt{1 - x^2} + x^2}{\sqrt{1 - x^2}}}{1 - x^2}$

Simplify this to:

$\frac{\sqrt{1 - x^x} \cdot \sqrt{1 - x^2} + x^2}{\sqrt{1 - x^2} \cdot (1 - x^2)}$

Final Expression:

The expression simplifies to:

$\frac{\sqrt{1 - x^x} \cdot \sqrt{1 - x^2} + x^2}{(1 - x^2)^{3/2}}$

---

Would you like me to explain any of the steps in more detail or clarify something further? Here are some related questions:

1. How can we approach simplifying expressions with nested powers like $x^x$?
2. What is the significance of the square root and inverse square root terms in simplifying algebraic expressions?
3. How does simplifying rational expressions with square roots differ from regular algebraic fractions?
4. How can we handle denominators involving square roots in expressions?
5. What are some strategies to handle complex expressions when both algebra and functions are involved?

Tip: Always look for common denominators when simplifying expressions with multiple fractions in the numerator or denominator!