24. (1 + 𝑖)^10 નો વાƨતિવક ભાગ શોધો.

પ્રશ્ન છે: \( (1 + i)^{10} \) નો વાસ્તવિક ભાગ શોધો.

### સમાધાન:
પ્રથમ, \( 1 + i \) ને ધ્રુવીય સ્વરૂપમાં રૂપાંતરિત કરીએ. 

1. **સત્તાધી રૂપ** (Polar form) માટે \( 1 + i \):

   - **માપ** (Magnitude): 
     \[
     r = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}
     \]
   
   - **કોણ** (Argument, \( \theta \)): 
     \[
     \theta = \tan^{-1} \left( \frac{1}{1} \right) = \frac{\pi}{4}
     \]
     
   તેથી \( 1 + i \) નો ધ્રુવીય રૂપ છે:
   \[
   1 + i = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)
   \]

2. **દશમા ઘાત માટે**:

   હવે \( (1 + i)^{10} \) નું ધ્રુવીય સ્વરૂપ કાઢીએ:
   \[
   (1 + i)^{10} = \left( \sqrt{2} \right)^{10} \left( \cos \left( 10 \cdot \frac{\pi}{4} \right) + i \sin \left( 10 \cdot \frac{\pi}{4} \right) \right)
   \]
   
   - \( \left( \sqrt{2} \right)^{10} = 2^5 = 32 \)
   - \( 10 \cdot \frac{\pi}{4} = \frac{10\pi}{4} = \frac{5\pi}{2} \)

   હવે \( \frac{5\pi}{2} \) ને \( 2\pi \) ના ગુણકમાં લાવીશું:
   \[
   \frac{5\pi}{2} = 2\pi + \frac{\pi}{2}
   \]
   તેથી:
   \[
   \cos \frac{5\pi}{2} = \cos \frac{\pi}{2} = 0
   \]
   \[
   \sin \frac{5\pi}{2} = \sin \frac{\pi}{2} = 1
   \]

   હવે આ મૂલ્યને મૂકીએ:
   \[
   (1 + i)^{10} = 32 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) = 32 (0 + i \cdot 1) = 32i
   \]

3. **વાસ્તવિક ભાગ**: 
   \( 32i \) માં વાસ્તવિક ભાગ છે \( 0 \).

### અંતિમ ઉત્તર:
\( (1 + i)^{10} \) નો વાસ્તવિક ભાગ **0** છે.

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Do you want more details or have any questions?

### Additional Questions:
1. What is the general method to find the imaginary part of a complex number raised to a power?
2. How can De Moivre's Theorem be used in complex number calculations?
3. Why does the real part of \( (1 + i)^{10} \) turn out to be zero?
4. What is the geometric interpretation of raising a complex number to a power?
5. How would the result differ if we were asked for the imaginary part instead?

#### Tip: Always simplify the angle in terms of \(2\pi\) to make calculations easier!

24. (1 + 𝑖)^10 નો વાસ્તવિક ભાગ શોધો.

Discover how to find the real part of (1 + i)^10 by converting the complex number into its polar form and applying De Moivre's Theorem. This problem explores key concepts of complex numbers, polar form, and exponentiation, suitable for advanced high school or college-level students.

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