Write the expression in the standard form a + bi.
-$$( 1 + i ) ^ { 20 }$$
A) $1024 \mathrm { i }$
B) 1024
C) $- 1024 \mathrm { i }$
D) $- 1024$

Question

Quizplus · Accepted Answer

To express $(1 + i)^{20}$ in the standard form $a + bi$, we first simplify the base $1 + i$. By using the formula $e^{i\theta} = \cos(\theta) + i\sin(\theta)$, we can express $1 + i$ in polar form. However, a more straightforward approach for this specific problem involves recognizing that $1 + i$ can be simplified using its magnitude and angle. The magnitude of $1 + i$ is $\sqrt{1^2 + 1^2} = \sqrt{2}$, and it forms a 45-degree angle with the real axis, which is $\pi/4$ radians. Therefore, $1 + i = \sqrt{2}(\cos(\pi/4) + i\sin(\pi/4))$. When raised to the 20th power, we focus on the exponential effect on both the magnitude and the angle: the magnitude $(\sqrt{2})^{20} = 2^{10} = 1024$, and the angle $20 \times \pi/4 = 5\pi$, which is equivalent to $\pi$ radians (or 180 degrees) modulo $2\pi$. This angle corresponds to a point on the negative real axis in the complex plane, meaning the imaginary part is 0. Therefore, $(1 + i)^{20}$ simplifies to $-1024$, which is purely real and matches option D.

Write the Expression in the Standard Form a + Bi (1+i)20( 1 + i ) ^ { 20 }(1+i)20

Write the Expression in the Standard Form a + Bi $( 1 + i ) ^ { 20 }$