# The 5 th power of the complex number ( 2 3 + 2 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 35E
To determine

## To find: The 5th power of the complex number (23+2i) by using De Moivre’s Theorem.

Expert Solution

The value of the complex number (23+2i)5 is 5123+512i.

### 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 (23+2i) 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 (23+2i).

Obtain the argument of the complex number (23+2i).

tanθ=223=13

Thus, the argument of argument of the complex number (23+2i) is θ=tan1(13)=π6

Obtain the modulus of the complex number (23+2i).

r=|(23+2i)|==(23)2+(2)2=12+4=4

Thus, the value of r=4.

Therefore, the polar form of the complex number (23+2i) is (23+2i)=4(cosπ6+isinπ6).

Simplify the expression (23+2i)5 by the use of De Moivre’s Theorem.

(23+2i)5=(4(cosπ6+isinπ6))5=(4)5[cos(5π6)+isin(5π6)]=1024(32+12i)=5123+512i

Thus, the value of the complex number (13i)5 is 5123+512i.

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