# The 5 th power of the complex number ( 1 − 3 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 34E
To determine

## To find: The 5th power of the complex number (1−3i) by using De Moivre’s Theorem.

Expert Solution

The value of the complex number (13i)5 is 16+163i.

### 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 (13i) 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 13i.

Obtain the argument of the complex number 13i.

tanθ=ba=31=3

Thus, the argument of argument of the complex number 13i is θ=tan1(3)=5π3

Obtain the modulus of the complex number 13i.

r=|13i|==12+(3)2=1+3=2

Thus, the value of r=2.

Therefore, the polar form of the complex number 13i is 13i=2(cos5π3+isin5π3).

Simplify the expression (13i)5 by the use of De Moivre’s Theorem.

(13i)5=(2(cos5π3+isin5π3))5=(2)5[cos(5(5π3))+isin(5(5π3))]=25(cos25π3+isin25π3)=32(12+i(32))=16+163i

Thus, the value of the complex number (13i)5 is 16+163i.

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