# The 8 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 36E
To determine

## To find: The 8th power of the complex number (1−i) by using De Moivre’s Theorem.

Expert Solution

The value of the complex number (1i)8 is 16.

### 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 (1i) 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.

Consider the complex number (1i).

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

tanθ=11=1

Thus, the argument of argument of the complex number (1i) is θ=tan1(1)=7π4

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

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

Thus, the value of r=2.

Therefore, the polar form of the complex number (1i) is (1i)=2(cos7π4+isin7π4).

Simplify the expression (1i)8 by the use of De Moivre’s Theorem.

(1i)8=(2(cos7π4+isin7π4))5=(2)8[cos((7π4)8)+isin((7π4)8)]=24(cos14π+isin14π)=16(1+0i)=16

Thus, the value of the complex number (13i)5 is 16.

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