To solve these problems, we will use Euler's formula, which is given by:

$e^{ix} = \cos(x) + i \sin(x)$

a. $\exp(2 + 3i)$

This expression is equivalent to $e^{2 + 3i}$, and can be separated into two parts:

$e^{2 + 3i} = e^2 \cdot e^{3i}$

Using Euler's formula for $e^{3i}$:

$e^{3i} = \cos(3) + i \sin(3)$

Thus, the full expression becomes:

$e^{2 + 3i} = e^2 \cdot (\cos(3) + i \sin(3))$

Now, we calculate the real and imaginary parts:

$\text{Real part: } e^2 \cos(3)$ $\text{Imaginary part: } e^2 \sin(3)$

Therefore, $\exp(2 + 3i) = e^2 \cos(3) + e^2 \sin(3) i$.

b. $e^{\left(\frac{3\pi}{2}i\right)}$

Using Euler's formula directly for $e^{ix}$, we have:

$e^{\left(\frac{3\pi}{2}i\right)} = \cos\left(\frac{3\pi}{2}\right) + i \sin\left(\frac{3\pi}{2}\right)$

From trigonometric values:

$\cos\left(\frac{3\pi}{2}\right) = 0, \quad \sin\left(\frac{3\pi}{2}\right) = -1$

Thus:

$e^{\left(\frac{3\pi}{2}i\right)} = 0 - i$

So the expression becomes $0 - i$.

c. $9^{6 - 5i}$

We can rewrite $9^{6 - 5i}$ as:

$9^{6 - 5i} = (9^6) \cdot 9^{-5i}$

Since $9 = e^{\ln(9)}$, we convert $9^{-5i}$ to an exponential form:

$9^{-5i} = e^{-5i \ln(9)}$

Using Euler's formula:

$e^{-5i \ln(9)} = \cos(5 \ln(9)) + i \sin(5 \ln(9))$

Thus, the full expression becomes:

$9^{6 - 5i} = 9^6 \cdot (\cos(5 \ln(9)) + i \sin(5 \ln(9)))$

Finally, the real part is $9^6 \cos(5 \ln(9))$ and the imaginary part is $9^6 \sin(5 \ln(9)) i$.

Would you like a more detailed breakdown or further clarifications on any part of the solution?

Related questions:

1. What is the general form of Euler's formula?
2. How do we split complex exponentials into real and imaginary parts?
3. What are the trigonometric values of angles like $3\pi/2$?
4. How do logarithms help in expressing powers of numbers in complex exponentials?
5. Can Euler's formula be extended to solve differential equations involving complex numbers?

Tip:

For exponentials of complex numbers, always try to break the problem into the product of a real exponential and a complex exponential using Euler's formula.