To find the first three terms in the expansion of \((1 - x)^{\frac{1}{4}}\), we can use the **binomial series expansion**. The binomial series for \((1 + x)^n\) is given by:

\[
(1 + x)^n = 1 + nx + \frac{n(n - 1)}{2!}x^2 + \frac{n(n - 1)(n - 2)}{3!}x^3 + \cdots
\]

For \((1 - x)^{\frac{1}{4}}\), let \(n = \frac{1}{4}\) and replace \(x\) by \(-x\) in the expansion:

\[
(1 - x)^{\frac{1}{4}} = 1 + \frac{1}{4}(-x) + \frac{\frac{1}{4} \left( \frac{1}{4} - 1 \right)}{2!}(-x)^2 + \cdots
\]

Now let's calculate the first three terms:

1. **First term**: \( 1 \)

2. **Second term**:
   \[
   \frac{1}{4} \cdot (-x) = -\frac{x}{4}
   \]

3. **Third term**:
   \[
   \frac{\frac{1}{4} \left( \frac{1}{4} - 1 \right)}{2!}(-x)^2 = \frac{\frac{1}{4} \cdot \left( -\frac{3}{4} \right)}{2}x^2 = -\frac{3}{32}x^2
   \]

Thus, the first three terms of the expansion are:

\[
(1 - x)^{\frac{1}{4}} \approx 1 - \frac{x}{4} - \frac{3}{32}x^2
\]

Would you like more details on any step?

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Here are 5 questions to explore further:

1. How does the binomial series generalize for any exponent \(n\)?
2. What are the conditions for the convergence of the binomial series?
3. How does this series behave for large values of \(x\)?
4. How can you find higher-order terms in this series?
5. How does this expansion change for different values of the exponent \(n\)?

**Tip:** The binomial series is especially useful when expanding expressions involving small values of \(x\), allowing for approximate calculations.

Learn how to find the first three terms of the binomial expansion of (1 - x)^(1/4) using the binomial series formula. This detailed solution explores binomial expansions, exponents, and series expansions, suitable for students in Grades 10-12.

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