Thank you for the clarification. Let's address the problem, focusing on the expansion of $(1 + 5x)^{3/5}$ in ascending powers of $x$ up to and including the term in $x^2$. Here's how you would approach each part of the problem:

Part (a): Expansion of $(1 + 5x)^{3/5}$

To expand $(1 + 5x)^{3/5}$ using the binomial theorem, we use the general formula for the binomial expansion:

$(1 + ax)^n = 1 + n(ax) + \frac{n(n-1)}{2!}(ax)^2 + \frac{n(n-1)(n-2)}{3!}(ax)^3 + \cdots$

Here, $n = \frac{3}{5}$ and $a = 5$. We expand up to the term in $x^2$:

$(1 + 5x)^{3/5} = 1 + \frac{3}{5}(5x) + \frac{\frac{3}{5} \left(\frac{3}{5} - 1\right)}{2!} (5x)^2 + \cdots$

Simplifying the expression:

$= 1 + 3x + \frac{\frac{3}{5} \left(\frac{-2}{5}\right)}{2} \cdot 25x^2$

$= 1 + 3x - \frac{6}{5} \times 5x^2$

$= 1 + 3x - \frac{6}{5} \times 5x^2$

$= 1 + 3x - \frac{6}{5} \times 25x^2 = 1 + 3x - 6x^2$

So the expansion up to $x^2$ is:

$(1 + 5x)^{3/5} = 1 + 3x - 6x^2$

Part (b): Approximate Value

Now, to use the expansion to find an approximate value for a function at a specific value of $x$, you would substitute the value of $x$ into the expansion. For example, if you needed to find the approximate value of $(1 + 5x)^{3/5}$ at $x = 0.01$, you would substitute $x = 0.01$ into the expression:

$1 + 3(0.01) - 6(0.01)^2 = 1 + 0.03 - 0.0006 = 1.0294$

Part (c): Percentage Error

If you have a specific actual value of the function, say at $x = 0.01$, you would compare it to the approximation found in part (b) to calculate the percentage error:

$\text{Percentage Error} = \left| \frac{\text{Actual Value} - \text{Approximate Value}}{\text{Actual Value}} \right| \times 100\%$

For example, if the actual value of $(1 + 5 \times 0.01)^{3/5}$ is known, you can plug it in to find the percentage error.

Would you like further details or have any specific values you'd like to evaluate in part (b) or (c)?

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Here are some related questions to deepen understanding:

1. How is the binomial expansion formula derived for fractional exponents?
2. What are the limitations of using a truncated binomial expansion for approximation?
3. Why does the expansion yield better approximations when $x$ is small?
4. How does the sign of the coefficient in the expansion affect the approximation?
5. What happens if we expand $(1 + 5x)^{3/5}$ beyond the $x^2$ term?
6. Can the binomial expansion be used for negative values of $x$ in this problem?
7. How does the expansion change if $x$ is a complex number?
8. What role does the binomial expansion play in solving differential equations?

Tip: Always consider the error introduced by truncating a series expansion. Higher-order terms can have significant effects, especially for larger values of $x$.