Question

The real part of $e^{(3+4i)x}$ is

• A. $e^{3x}$
• B. $\cos 7x$
• C. $e^{3x} \cos 4x$
• D. $e^{3x} \sin 4x$
• E. $0$

Hint:

Complex Numbers Tip: Whenever you see $e^{(a+bi)x}$, it will always expand to a real part of $e^{ax} \cos(bx)$ and an imaginary part of $e^{ax} \sin(bx)$.

Answer 1

The Correct Option is C

Concept:
Complex exponential expressions can be expanded using Euler's formula, which states that $e^{i\theta} = \cos \theta + i \sin \theta$. The expression must first be separated into its real and imaginary exponents.

Step 1: Write the initial expression.
The given complex exponential function is: $$Z = e^{(3+4i)x}$$

Step 2: Distribute the variable x.
Multiply the $x$ into the terms inside the parentheses: $$Z = e^{3x + 4ix}$$

Step 3: Apply exponential rules.
Use the exponent addition rule $e^{a+b} = e^a \cdot e^b$ to split the real and imaginary parts of the exponent: $$Z = e^{3x} \cdot e^{i(4x)}$$

Step 4: Apply Euler's formula.
Use Euler's formula $e^{i\theta} = \cos \theta + i \sin \theta$, where $\theta = 4x$: $$Z = e^{3x} (\cos 4x + i \sin 4x)$$ $$Z = e^{3x} \cos 4x + i e^{3x} \sin 4x$$

Step 5: Identify the real part.
A complex number is written in the form $a + ib$, where $a$ is the real part. Here, the real part (the term without $i$) is: $$\text{Real}(Z) = e^{3x} \cos 4x$$ Hence the correct answer is (C) $e^{3x \cos 4x$}.