Q4P - Mathematical Methods in Physical Sciences

Question

Find each of the following in rectangular form $\mathbf{x}\mathbf{+}\mathbf{iy}$ and check your results by computer. Remember to save time by doing as much as you can in your head.

${\mathit{e}}^{\mathbf{3}\mathbf{l}\mathbf{n}\mathbf{-}\mathit{i}\mathit{\pi }}$.

Step-by-Step Solution

Answer

The rectangular form of the given question is $e^{3\ln 2-i\pi }=-8$.

Step 1: Given Information.

The given expression is $e^{3\ln 2-i\pi }$.

Step 2: Meaning of rectangular form.

Representing the complex number in rectangular form means writing the given complex number in the form of $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$, in which x is the real part and y is the imaginary part.

Step 3: Separate the exponential.

The given question is $e^{3\ln 2-i\pi }$

 

Break the exponential part in the given question.

$$\begin{gathered} e^{3\ln 2-i\pi }=e^{3\ln 2}{e}^{-i\pi } \\ {e}^{3\ln 2-i\pi }=e^{\ln 2^3}{e}^{-i\pi } \\ {e}^{3\ln 2-i\pi }=e^{\ln 8}{e}^{-i\pi } \\ {e}^{3\ln 2-i\pi }=8e^{-i\pi } \end{gathered}$$

Step 4: Convert it into rectangular form.

Use the complex number formula $e^{-i\theta }=\cos \theta -i \sin \theta$ to rewrite the above expression.

$$\begin{gathered} 8e^{-i\pi }=8\left(\cos \pi -i \sin \pi \right) \\ =8(-1-i(0\left)\right) \\ =-8 \end{gathered}$$

 

Therefore, the rectangular form of $e^{3\ln 2-i\pi }$ is -8.