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The question in the image reads:

**"If x be so small that its square and higher powers may be neglected, then evaluate"** the following expression:

\[
\frac{(1+x)^n}{(9+2x)^{1/2}}
\]

To solve this, let's apply the approximation method that takes advantage of the fact that \( x \) is very small. This means we can neglect the terms involving \( x^2 \) and higher powers of \( x \).

### Step 1: Approximate the expression for small \( x \)

We start by approximating both the numerator and denominator for small \( x \).

#### Numerator: \( (1+x)^n \)
For small \( x \), we can use the binomial approximation:
\[
(1+x)^n \approx 1 + nx \quad \text{(neglecting higher powers of \( x \))}
\]

#### Denominator: \( (9+2x)^{1/2} \)
We can expand \( (9+2x)^{1/2} \) using the binomial approximation for fractional powers:
\[
(9+2x)^{1/2} = 9^{1/2} \left( 1 + \frac{2x}{9} \right)^{1/2}
\]
For small \( x \), we approximate this as:
\[
(9+2x)^{1/2} \approx 3 \left( 1 + \frac{x}{9} \right) \approx 3 + \frac{x}{3}
\]

### Step 2: Putting it together

Now substitute the approximations into the original expression:
\[
\frac{(1+x)^n}{(9+2x)^{1/2}} \approx \frac{1 + nx}{3 + \frac{x}{3}}
\]

### Step 3: Simplify the expression

To simplify the expression, we can use the approximation \( 3 + \frac{x}{3} \approx 3 \) for small \( x \), which leads to:
\[
\frac{1 + nx}{3} = \frac{1}{3} + \frac{nx}{3}
\]

Thus, the simplified result is approximately:
\[
\frac{1}{3} + \frac{nx}{3}
\]

### Final Answer:
The approximate value of the expression is:
\[
\frac{1}{3} + \frac{nx}{3}
\]

Would you like further clarification or any details on this solution? Here are a few related questions:

1. What is the binomial approximation for \( (1+x)^n \) when \( x \) is small?
2. How do you apply the binomial expansion for fractional powers like \( (9+2x)^{1/2} \)?
3. How can this approximation method be generalized for other types of expressions involving small variables?
4. What happens if higher-order terms of \( x \) are not neglected? How would the result change?
5. What is the role of neglecting higher powers of \( x \) in simplifying expressions?

**Tip:** When approximating for small values of \( x \), it is crucial to only keep the first few terms of the expansion and discard higher-order terms to make the calculation manageable and quick.

If x be so small that its square and higher powers may be neglected, then evaluate (1+x)^n / (9+2x)^(1/2).

Learn how to simplify the expression (1+x)^n / (9+2x)^(1/2) using the binomial approximation method for small x. This involves neglecting higher powers of x to evaluate the function. Suitable for students in Grades 11-12 studying binomial expansion.

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