Question

Find each of the following in rectangular form $x+iy$. $$e^{3\ln 2-i\pi }$$

Short answer

-8

Step-by-step solution

Simplify the exponent terms

First, solve the exponents in the given expression. Note that $e^{3\ln 2}$ can be rewritten. Since $e^{\ln a}=a{e}^{k\ln a}=a^k$, we get: $$e^{3\ln 2}=2^3=8$$

Combine simplified terms

Combine the expression after simplifying the exponents: $$e^{3\text{(}\ln 2\text{)}-i\pi }=e^{8-i\pi }$$

Apply Euler's formula

Utilize Euler's formula, $e^{i\theta }=\cos \theta +i\sin \theta$, to convert the complex exponential form to rectangular form: $$e^{8-i\pi }=8(e^{-i\pi })$$

Evaluate using trigonometric identities

Substitute $\pi$ into Euler's formula, knowing $e^{-i\pi }=\cos (-\pi )+i\sin (-\pi )=\text{-}1$: $$8(-1)=-8$$

Key concepts

Rectangular Form

Rectangular form, also known as Cartesian form, is the representation of complex numbers using the format: $$x+iy$$ Here, **x** is the real part, and **i y** is the imaginary part of the complex number. It combines both components in one expression, making it very versatile for different types of calculations.
To convert any complex number in exponential form to rectangular form, we can use Euler's formula and various trigonometric identities. This showcases the beauty of viewing complex numbers from different perspectives.

Euler's Formula

Euler's formula is instrumental in connecting the exponential form of complex numbers to their trigonometric form. The formula is given by:$$e^{i\theta }=\text{cos}\theta +i\text{sin}\theta$$Here, **θ** is the angle in radians, which the complex number makes with the positive x-axis. This relationship helps simplify many complex calculations, including converting between different forms of complex numbers.
Using Euler's formula, we can easily change exponential forms into their corresponding rectangular forms, and it opens up a way to use trigonometric functions in complex number operations.

Trigonometric Identities

Trigonometric identities play a crucial role in working with complex numbers, especially when converting between forms. Some important identities to remember are:

• $$\text{cos}(-\theta )=\text{cos}(\theta )$$ and $$\text{sin}(-\theta )=-\text{sin}(\theta )$$
• $$\text{cos}(\theta +2\text{π})=\text{cos}(\theta )$$ and $$\text{sin}(\theta +2\text{π})=\text{sin}(\theta )$$

In the context of complex numbers:

• $$e^{i\text{π}}=\text{cos}(\text{π})+i\text{sin}(\text{π})=-1$$ since $$\text{cos}(\text{π})=-1$$ and $$\text{sin}(\text{π})=0$$

These identities help streamline the process of converting from complex exponential form to rectangular form.

Complex Exponential

The complex exponential form is a powerful way to represent complex numbers. It is expressed as:$$r{e}^{i\theta }$$where **r** is the modulus (or magnitude) of the complex number and **θ** is the argument (or angle in radians). This form is particularly useful in various fields such as physics, engineering, and applied mathematics.
In our example, starting with $$e^{3\text{ln}2-i\text{π}}$$we first simplify the term:$$e^{3\text{ln}2}=2^3=8$$Then we use Euler's formula:$$e^{-i\text{π}}=\text{cos}(-\text{π})+i\text{sin}(-\text{π})$$which simplifies further to $$-1$$So, multiplying by 8, we get the rectangular form: $$-8$$