The image shows a mathematics problem related to the expansion of (1+5x)^3/5 in ascending powers of x*x* up to and including the term in x2*x*2. The problem is divided into three parts:

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 \(x\).

Explore the expansion of (1 + 5x)^(3/5) in ascending powers of x up to and including x^2. Learn how to apply the binomial theorem for fractional exponents and approximate values. Suitable for advanced high school students.

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