Visual SolutionPYQ 2024Standard

Question

The value of

(1 + i)^8

is:

(A)

16

(B)

-16

(C)

16i

(D)

-16i

Solution Path

Convert

1+i

to polar:

\sqrt{2}\, e^{i\pi/4}

. Raise to 8th power:

(\sqrt{2})^8 = 16

e^{i \cdot 2\pi} = 1

. Answer:

16

01Question Setup

1/4

Find the value of

(1+i)^8

(1+i)^8 = \,?

02Polar Form

2/4

Convert

1+i

to polar form: modulus

\sqrt{2}

, argument

\pi/4

1+i = \sqrt{2}\,e^{i\pi/4}

03Apply PowerKEY INSIGHT

3/4

Raise modulus to the 8th power and multiply the argument by 8.

(\sqrt{2})^8 = 16

and

e^{i \cdot 2\pi} = 1

(\sqrt{2})^8 = 16,\; e^{i \cdot 2\pi} = 1

04Final Answer

4/4

The result is a real number:

16 \times 1 = 16

\boxed{(1+i)^8 = 16}

Concepts from this question2 concepts unlocked

★

STANDARD

(cos \theta + i sin \theta)^n = cos n\theta + i sin n\theta

(e^{i\theta})^n = e^{in\theta}

Powers of complex numbers become trivial. Core tool for roots of unity problems.

nth rootsPowers of complex numbersTrigonometric identities

EASY

Multiplying in Euler form = adding exponents

e^{i\alpha} \cdot e^{i\beta} = e^{i(\alpha + \beta)}

Turns complex multiplication into simple addition - saves 2-3 minutes per question

Roots of unityDe Moivre's applicationsProduct/sum problems

Want more practice?