# The 20 th power of the complex number ( 1 + i ) by using De Moivre’s Theorem.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter I, Problem 33E
To determine

## To find: The 20th power of the complex number (1+i) by using De Moivre’s Theorem.

Expert Solution

The value of the complex number (1+i)20 is 1024.

### Explanation of Solution

Theorem used:

De Moivre’s Theorem:

If z=r(cosθ+isinθ) and n be a positive integer then,

zn=[r(cosθ+isinθ)]n=rn(cosnθ+isinnθ).

Calculation:

Rewrite the complex number (1+i) in polar form.

The polar form of the complex number z=a+bi is z=r(cosθ+isinθ) where r=|z|=a2+b2  and tanθ=ba.

Obtain the argument of the complex number (1+i).

tanθ=ba=11=1

Thus, the argument of argument of the complex number (1+i) is θ=tan1(1)=π4

Obtain the modulus of the complex number (1+i).

r=|1+i|==12+12=2

Thus, the value of r=2.

Therefore, the polar form of the complex number (1+i) is (1+i)=2(cosπ4+isinπ4).

Simplify the expression (1+i)20 by the use of De Moivre’s Theorem.

(1+i)20=(2(cosπ4+isinπ4))20=(2)20[cos(20(π4))+isin(20(π4))]=210(cos5π+isin5π)=210(1+i(0))=1024

Thus, the value of the complex number (1+i)20 is 1024.

### Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!