We are tasked with simplifying the expression:

$(1 - x)^{-5/2} + \sqrt{16 + 8x} \div \left( (1 + x)^{-1/2} (2 + x)^2 \right)$

Given that $x$ is small, we will approximate the expression using a Taylor expansion for small $x$.

Step 1: Approximation of $(1 - x)^{-5/2}$

For small $x$, the binomial expansion of $(1 - x)^{n}$ is:

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

For $(1 - x)^{-5/2}$, $n = -5/2$:

$(1 - x)^{-5/2} \approx 1 + \frac{-5}{2}x + \frac{\frac{-5}{2} \cdot \frac{-7}{2}}{2} x^2 = 1 - \frac{5}{2}x + \frac{35}{8}x^2$

Step 2: Approximation of $\sqrt{16 + 8x}$

Next, we expand $\sqrt{16 + 8x}$ around $x = 0$. Factor out the constant:

$\sqrt{16 + 8x} = 4 \sqrt{1 + \frac{x}{2}}$

For small $x$, use the binomial expansion for $\sqrt{1 + u}$:

$\sqrt{1 + u} \approx 1 + \frac{u}{2}$

Substituting $u = \frac{x}{2}$:

$\sqrt{16 + 8x} = 4 \left( 1 + \frac{x}{4} \right) = 4 + x$

Step 3: Approximation of $(1 + x)^{-1/2}$

Similarly, for small $x$, expand $(1 + x)^{-1/2}$:

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

Step 4: Approximation of $(2 + x)^2$

Expand $(2 + x)^2$:

$(2 + x)^2 = 4 + 4x + x^2$

Step 5: Putting everything together

Now, substitute the approximations into the expression:

$\frac{\sqrt{16 + 8x}}{(1 + x)^{-1/2} (2 + x)^2} \approx \frac{4 + x}{(1 - \frac{x}{2})(4 + 4x + x^2)}$

Approximate the denominator for small $x$:

$(1 - \frac{x}{2})(4 + 4x + x^2) \approx (4 - 2x)(4 + 4x) \approx 16 + 16x - 8x - 4x^2 = 16 + 8x - 4x^2$

Thus:

$\frac{4 + x}{16 + 8x - 4x^2} \approx \frac{4 + x}{16(1 + \frac{x}{2} - \frac{x^2}{4})} \approx \frac{4 + x}{16} (1 - \frac{x}{2} + \frac{x^2}{4})$

$\approx \frac{4}{16} (1 - \frac{x}{2} + \frac{x^2}{4}) + \frac{x}{16} (1 - \frac{x}{2})$

$= \frac{1}{4} - \frac{x}{8} + \frac{x^2}{16} + \frac{x}{16} - \frac{x^2}{32}$

Combining like terms:

$= \frac{1}{4} - \frac{x}{16} + \frac{x^2}{32}$

Final expression

Adding $(1 - x)^{-5/2} \approx 1 - \frac{5}{2}x + \frac{35}{8}x^2$ to this:

$1 - \frac{5}{2}x + \frac{35}{8}x^2 + \left( \frac{1}{4} - \frac{x}{16} + \frac{x^2}{32} \right)$

$= 1 + \frac{1}{4} - \frac{5}{2}x - \frac{x}{16} + \frac{35}{8}x^2 + \frac{x^2}{32}$

Thus, for small $x$, the simplified expression is approximately:

$\frac{5}{4} - \frac{41}{16}x + \frac{145}{32}x^2$

Would you like more details on this or have any questions?

Here are five related questions:

1. How does the binomial expansion help in approximations for small values of $x$?
2. Can you explain the general form of Taylor series used in approximations?
3. Why do we factor out constants when approximating functions like $\sqrt{16 + 8x}$?
4. How do higher-order terms affect the accuracy of small $x$ approximations?
5. Can you expand $(1 - x)^{-3/2}$ for small $x$ using binomial expansion?

Tip: When simplifying expressions with small variables, keeping up to the second-order terms often provides a good balance between accuracy and simplicity.