Let's analyze the expression step by step:

Expression Breakdown:

The first part of the equation given is:

$\frac{1}{2} \cdot \frac{(x^2 - 1)^{-\frac{1}{2}}}{(x^2 + 1)^{-\frac{1}{2}}} \cdot \frac{4x}{(x^2 + 1)^2}$

Step-by-Step Simplification:

1. Combine Terms: $\frac{1}{2} \cdot \frac{(x^2 - 1)^{-\frac{1}{2}}}{(x^2 + 1)^{-\frac{1}{2}}} \cdot \frac{4x}{(x^2 + 1)^2}$

   • We see two separate fractions here, and the first task is to multiply the fractions together. This leads to: $\frac{1}{2} \cdot \frac{4x (x^2 - 1)^{-\frac{1}{2}}}{(x^2 + 1)^{-\frac{1}{2}} (x^2 + 1)^2}$

2. Simplify the constants: $\frac{1}{2} \cdot 4x = 2x$ So now the equation becomes: $\frac{2x (x^2 - 1)^{-\frac{1}{2}}}{(x^2 + 1)^{-\frac{1}{2}} (x^2 + 1)^2}$

3. Combine the powers of $(x^2 + 1)$: The denominator has two factors involving $(x^2 + 1)$: $(x^2 + 1)^{-\frac{1}{2}} \cdot (x^2 + 1)^2$ Using the exponent rules (specifically, $a^m \cdot a^n = a^{m+n}$), we can add the exponents: $-\frac{1}{2} + 2 = \frac{3}{2}$ So the new expression becomes: $\frac{2x (x^2 - 1)^{-\frac{1}{2}}}{(x^2 + 1)^{\frac{3}{2}}}$

4. Final Simplified Form: Thus, the expression simplifies to: $\frac{2x}{(x^2 - 1)^{\frac{1}{2}} (x^2 + 1)^{\frac{3}{2}}}$

This matches the result shown in the second step of your provided image.

Let me know if you need more details, or if you have any questions! Here are 5 related questions that could help you explore further:

1. How would you approach simplifying complex fractions like this in general?
2. Can you explain the power rules used in this simplification?
3. What is the importance of exponent manipulation in simplifications involving radicals?
4. How could this problem be approached differently, for instance, if negative exponents were avoided from the start?
5. How do constants like $\frac{1}{2}$ and $4$ factor into simplifying algebraic expressions like this?

Tip:

When simplifying expressions with multiple exponents, always apply the rules of exponents (multiplication, division, and power of powers) carefully to avoid mistakes.