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To solve this problem, let's break it down step-by-step.

### Step 1: Expand \( (1 + 2x)^{10} \) up to the fourth term in powers of \( x \).

This is a binomial expansion problem where we expand \( (1 + 2x)^{10} \) using the Binomial Theorem:
\[
(1 + 2x)^{10} = \sum_{k=0}^{10} \binom{10}{k} \cdot (1)^{10-k} \cdot (2x)^k
\]

We are asked to expand up to and including the fourth term. So, we need the terms for \( k = 0, 1, 2, 3, \) and \( 4 \).

#### Terms Calculation:
1. **For \( k = 0 \):**
   \[
   \binom{10}{0} \cdot (1)^{10} \cdot (2x)^0 = 1
   \]

2. **For \( k = 1 \):**
   \[
   \binom{10}{1} \cdot (1)^{9} \cdot (2x)^1 = 10 \cdot 2x = 20x
   \]

3. **For \( k = 2 \):**
   \[
   \binom{10}{2} \cdot (1)^{8} \cdot (2x)^2 = 45 \cdot 4x^2 = 180x^2
   \]

4. **For \( k = 3 \):**
   \[
   \binom{10}{3} \cdot (1)^{7} \cdot (2x)^3 = 120 \cdot 8x^3 = 960x^3
   \]

5. **For \( k = 4 \):**
   \[
   \binom{10}{4} \cdot (1)^{6} \cdot (2x)^4 = 210 \cdot 16x^4 = 3360x^4
   \]

Thus, the expansion of \( (1 + 2x)^{10} \) up to the fourth term is:
\[
1 + 20x + 180x^2 + 960x^3 + 3360x^4
\]

### Step 2: Use the expansion to evaluate \( (1.0002)^{10} \)

To find \( (1.0002)^{10} \), we let \( x = 0.0001 \) in our expansion \( (1 + 2x)^{10} \).

Substitute \( x = 0.0001 \) into the expression:
\[
1 + 20(0.0001) + 180(0.0001)^2 + 960(0.0001)^3 + 3360(0.0001)^4
\]

Now, calculate each term:
1. \( 20 \times 0.0001 = 0.002 \)
2. \( 180 \times (0.0001)^2 = 180 \times 0.00000001 = 0.0000018 \)
3. \( 960 \times (0.0001)^3 = 960 \times 0.000000000001 = 0.000000096 \)
4. \( 3360 \times (0.0001)^4 = 3360 \times 0.0000000000000001 = 0.000000000336 \)

Add these terms together:
\[
1 + 0.002 + 0.0000018 + 0.000000096 + 0.000000000336 \approx 1.0020019
\]

### Final Answer:
To four decimal places, \( (1.0002)^{10} \approx 1.0020 \).

---

Would you like a detailed explanation of any specific part? Here are some related questions that might expand your understanding:

1. How does the Binomial Theorem work in general for any expansion \( (a + b)^n \)?
2. What are binomial coefficients, and how are they calculated?
3. Can the expansion be used for other values of \( n \) or different terms in the series?
4. What are some real-world applications of binomial expansion?
5. Why do we approximate \( (1.0002)^{10} \) using the first few terms only?

**Tip:** When dealing with binomial expansions for small changes (like \( 1.0002 \)), the first few terms usually provide a very close approximation.

Expand (1 + 2x)^10 in ascending powers of x^4, up to and including the fourth term. Hence, evaluate (1.0002)^10 correct to four decimal places.

Expand (1 + 2x)^10 in ascending powers of x up to the fourth term and approximate (1.0002)^10 to four decimal places. Explore the use of binomial expansion and approximation techniques in this advanced math problem suitable for Grades 11-12.

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