BuyFind

Single Variable Calculus: Concepts...

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

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

Answer to Problem 33E

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.

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