If x is small (1-x)^-5/2+(16+8x)^1/2÷(1+x)^-1/2(2+x)^2

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.

Learn how to simplify the expression (1 - x)^-5/2 + sqrt(16 + 8x) ÷ ((1 + x)^-1/2 (2 + x)^2) for small values of x using binomial and Taylor expansions. This problem covers key concepts such as approximation techniques for small x, binomial expansion, and Taylor series. Suitable for advanced high school or early university students.

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