To find the linear approximation of \((0.99)^{10}\), we use the concept of linear approximation based on a function's tangent line at a nearby point where the calculation is simpler. 

### Step-by-Step Solution:
1. **Define the function:**
   Let \(f(x) = x^{10}\).

2. **Choose a point near \(x = 0.99\):**
   We select \(x = 1\) because it's close to \(0.99\) and simplifies calculations.

3. **Compute \(f(x)\) and its derivative at \(x = 1\):**
   - \(f(x) = x^{10}\)
   - \(f'(x) = 10x^9\)

   At \(x = 1\):
   - \(f(1) = 1^{10} = 1\)
   - \(f'(1) = 10 \cdot 1^9 = 10\)

4. **Write the linear approximation formula:**
   The linear approximation is:
   \[
   f(x) \approx f(a) + f'(a)(x - a)
   \]
   where \(a = 1\).

   Substituting the values:
   \[
   f(x) \approx 1 + 10(x - 1)
   \]

5. **Approximate \(f(0.99)\):**
   Substitute \(x = 0.99\):
   \[
   f(0.99) \approx 1 + 10(0.99 - 1)
   \]
   \[
   f(0.99) \approx 1 + 10(-0.01)
   \]
   \[
   f(0.99) \approx 1 - 0.1 = 0.9
   \]

### Final Answer:
\[
(0.99)^{10} \approx 0.9
\]

Would you like more details or assistance with this topic?

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### 5 Related Questions:
1. How can we improve the accuracy of this approximation using higher-order terms?
2. What are some practical applications of linear approximations in real-world problems?
3. How does the error in linear approximation vary with the distance from the point of tangency?
4. Can you compute the exact value of \((0.99)^{10}\) and compare it to the approximation?
5. How would the approximation change if we used a point closer to \(0.99\), like \(0.995\)?

### Tip:
Linear approximation is most accurate when the chosen point is close to the value you're approximating, so selecting the right point is crucial!

Learn how to approximate (0.99)^10 using the concept of linear approximation. This detailed guide covers key concepts such as derivatives, exponents, and the linear approximation formula. Perfect for high school students in Grades 11-12.

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