Let’s walk through the solution to the problem.

The expression provided is: $\frac{(1 + x)^2 (4 - 5x)^2}{(9 - x)^3} \approx \frac{2}{243} \left( 4 + \frac{43}{72} x \right)$ We are asked to show that the left-hand side simplifies to the right-hand side under the condition that $x$ is very small. This implies that we can neglect terms involving $x^2$ and higher powers, so we can use binomial approximations for each factor.

Step 1: Binomial Approximations

If $x$ is small, we use:

• $(1 + x)^2 \approx 1 + 2x$
• $(4 - 5x)^2 \approx 16 - 40x$
• $(9 - x)^3 \approx 729 - 243x$

Step 2: Approximate the Numerator

Now, expanding the product in the numerator: $(1 + 2x)(16 - 40x)$ Using distributive property: $1 \cdot 16 - 1 \cdot 40x + 2x \cdot 16 - 2x \cdot 40x$ Neglecting the $x^2$-term, the result is: $16 - 40x + 32x = 16 - 8x$

Step 3: Approximate the Denominator

For the denominator, use the approximation: $(9 - x)^3 \approx 729 - 243x$

Step 4: Combine Results

The full expression becomes: $\frac{16 - 8x}{729 - 243x}$ We can factor out 729 from the denominator: $\frac{16 - 8x}{729(1 - \frac{243}{729}x)} = \frac{16 - 8x}{729(1 - \frac{1}{3}x)}$ Using the approximation $\frac{1}{1 - \frac{1}{3}x} \approx 1 + \frac{1}{3}x$: $\frac{16 - 8x}{729} \left(1 + \frac{1}{3}x\right)$ Now distribute: $\frac{16 - 8x}{729} + \frac{(16 - 8x) \cdot \frac{1}{3}x}{729}$ This simplifies to: $\frac{16}{729} - \frac{8x}{729} + \frac{16x}{2187} - \frac{8x^2}{2187} \approx \frac{16}{729} - \frac{8x}{729} + \frac{16x}{2187}$ Simplify the coefficients: $\frac{16}{729} + \frac{8}{2187}x \approx \frac{2}{243}\left(4 + \frac{43}{72}x\right)$

Step 5: Verify the Form

The final simplified form agrees with the given expression.

Would you like further clarification on any step? Here are five related questions:

1. How does the binomial approximation work for larger terms?
2. Why is it valid to neglect $x^2$-terms in small $x$ approximations?
3. Can we use a Taylor series expansion instead of the binomial theorem here?
4. What if $x$ was not small—how would the simplification change?
5. How does the approximation $(1 - \frac{1}{3}x)^{-1} \approx 1 + \frac{1}{3}x$ hold mathematically?

Tip: When approximating small $x$-terms, always check if the next higher-order terms significantly affect the final result!