https://file.aiquickdraw.com/mathbot/file/1727855003122crs2tz6q.jpg

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.

Simplify the expression: (1 - x^2)^(1/2) + x^2(1 - x^2)^(-1/2) divided by 1 - x^2

This problem simplifies an algebraic expression involving exponents and square roots: (1 - x^2)^(1/2) + x^2(1 - x^2)^(-1/2) divided by (1 - x^2). It covers essential concepts like exponent rules and factorization, suitable for students in Grades 10-12.

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