We are asked to simplify or analyze the expression:

$(1 - x)^{-5/2} + \frac{(16 + 8x)^{1/2}}{(1 + x)^{-1/2}} \cdot (2 + x)^2$

Let’s carefully simplify each part, considering $x$ is small. When $x$ is small, we can make use of approximations based on the binomial series expansion.

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Step-by-Step Solution

1. Simplifying $(1 - x)^{-5/2}$ for small $x$:

Use the binomial expansion for small $x$: $(1 - x)^{n} \approx 1 + nx + \frac{n(n-1)}{2}x^2 + \dots$ Here $n = -5/2$, so: $(1 - x)^{-5/2} \approx 1 + \left(-\frac{5}{2}\right)(-x) + \frac{\left(-\frac{5}{2}\right)\left(-\frac{7}{2}\right)}{2}x^2 + \dots$ $(1 - x)^{-5/2} \approx 1 + \frac{5}{2}x + \frac{35}{8}x^2 + \dots$

2. Simplifying $(16 + 8x)^{1/2}$ for small $x$:

We can factor out 16: $(16 + 8x)^{1/2} = \sqrt{16(1 + \frac{1}{2}x)} = 4 \cdot (1 + \frac{1}{2}x)^{1/2}$ Now, use the binomial expansion again: $(1 + \frac{1}{2}x)^{1/2} \approx 1 + \frac{1}{2} \cdot \frac{1}{2}x = 1 + \frac{1}{4}x$ Thus: $(16 + 8x)^{1/2} \approx 4 \left(1 + \frac{1}{4}x\right) = 4 + x$

3. Simplifying $(1 + x)^{-1/2}$ for small $x$:

Using the binomial expansion for $(1 + x)^{-1/2}$: $(1 + x)^{-1/2} \approx 1 - \frac{1}{2}x + \dots$

4. Simplifying the product with $(2 + x)^2$:

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

5. Putting it all together:

Now, the full second term becomes: $\frac{(4 + x)}{1 - \frac{1}{2}x} \cdot (4 + 4x + x^2)$ For small $x$, use the approximation $\frac{1}{1 - \frac{1}{2}x} \approx 1 + \frac{1}{2}x$: $\frac{4 + x}{1 - \frac{1}{2}x} \approx (4 + x)(1 + \frac{1}{2}x) = 4 + x + 2x + \frac{1}{2}x^2 = 4 + 3x + \frac{1}{2}x^2$ Now multiply by $4 + 4x + x^2$: $(4 + 3x + \frac{1}{2}x^2)(4 + 4x + x^2)$ Expanding: $16 + 16x + 4x^2 + 12x + 12x^2 + \frac{3}{2}x^3 + 2x^2 + 2x^3 + \frac{1}{2}x^4$ Simplify the terms: $16 + 28x + 18x^2 + \frac{7}{2}x^3 + \frac{1}{2}x^4$

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Final Expression:

Now, combining the two terms, we get: $(1 - x)^{-5/2} + \left(16 + 28x + 18x^2 + \frac{7}{2}x^3 + \frac{1}{2}x^4\right)$ Using the earlier approximation for $(1 - x)^{-5/2}$: $1 + \frac{5}{2}x + \frac{35}{8}x^2 + \dots + 16 + 28x + 18x^2 + \frac{7}{2}x^3 + \frac{1}{2}x^4$

Simplify the combined terms: $17 + \left(\frac{5}{2} + 28\right)x + \left(\frac{35}{8} + 18\right)x^2 + \dots$ $17 + \frac{61}{2}x + \frac{179}{8}x^2 + \dots$

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Conclusion:

The expression simplifies to: $17 + \frac{61}{2}x + \frac{179}{8}x^2 + \dots$

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Do you want further details or have any questions?

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5 Related Questions:

1. What is the binomial expansion, and when is it used?
2. How does the approximation of small values work in calculus?
3. What are higher-order terms in Taylor series expansions?
4. How can we simplify complex algebraic fractions?
5. What are common uses of fractional exponents in physics and engineering?

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Tip:

When $x$ is small, binomial approximations are extremely useful for quick calculations. Remember to identify when higher-order terms are insignificant to avoid unnecessary complexity!