Calculate (2 + 3i)(−1 + 4i) and represent the answer in exponential form.

Sure! To calculate (2 + 3i)(−1 + 4i), we can use the distributive property of multiplication.

Step 1: Multiply the real parts: 2 * -1 = -2

Step 2: Multiply the imaginary parts: 3i * -1 = -3i

Step 3: Multiply the real part of the first number with the imaginary part of the second number: 2 * 4i = 8i

Step 4: Multiply the imaginary part of the first number with the imaginary part of the second number: 3i * 4i = 12i^2

Step 5: Simplify the result: 12i^2 = 12 * -1 = -12

Step 6: Combine the real and imaginary parts: -2 + (-3i) + 8i - 12 = -14 + 5i

So, (2 + 3i)(−1 + 4i) equals -14 + 5i.

To represent the answer in exponential form, we can use Euler's formula, which states that e^(ix) = cos(x) + isin(x).

In this case, the real part is -14 and the imaginary part is 5. So, the exponential form of the answer is:

e^(-14 + 5i) = e^(-14) * e^(5i)

Therefore, the answer in exponential form is e^(-14) * e^(5i).