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 36E
To determine

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

Expert Solution

Answer to Problem 36E

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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