The expression in the image is:

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

Let's simplify this step by step:

1. Rewrite the numerator: The numerator consists of two terms: $(1 - x^2)^{\frac{1}{2}} + x^2(1 - x^2)^{-\frac{1}{2}}$

   The first term is straightforward: $(1 - x^2)^{\frac{1}{2}}$.

   The second term can be rewritten as: $x^2(1 - x^2)^{-\frac{1}{2}}$ which is $x^2$ multiplied by the reciprocal of $(1 - x^2)^{\frac{1}{2}}$.

2. Factor the numerator: Both terms have powers of $(1 - x^2)$. Factor out $(1 - x^2)^{-\frac{1}{2}}$ from both terms in the numerator: $(1 - x^2)^{-\frac{1}{2}} \left( (1 - x^2) + x^2 \right)$ Simplifying the expression inside the parentheses: $(1 - x^2) + x^2 = 1$ So, the numerator becomes: $(1 - x^2)^{-\frac{1}{2}} \cdot 1 = (1 - x^2)^{-\frac{1}{2}}$

3. Final expression: Now, the expression simplifies to: $\frac{(1 - x^2)^{-\frac{1}{2}}}{1 - x^2}$ Using the property of exponents, we can rewrite this as: $(1 - x^2)^{-\frac{1}{2} - 1} = (1 - x^2)^{-\frac{3}{2}}$

Thus, the simplified form of the expression is: $(1 - x^2)^{-\frac{3}{2}}$

Would you like more details or have any questions about this simplification?

Here are 5 related questions you might want to explore:

1. How can we apply this result in integration or differentiation problems?
2. What is the domain of the simplified function $(1 - x^2)^{-\frac{3}{2}}$?
3. How would the expression change if the powers were different?
4. What happens if $x = \pm1$ in the original or simplified expression?
5. How would the simplification proceed if we had additional terms in the numerator?

Tip: When simplifying expressions involving powers and roots, always look for common factors to reduce complexity early in the process.