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Chapter 9: Q. 49 (page 592)

In Problems 41鈥�52, write each expression in the standard form a+bi.

49.(1-i)5

Short Answer

Expert verified

(1-i)5=-4+4i.

Step by step solution

01

Step 1. Rewrite 1-i in the form r(cos∅+isin∅).

The magnitude of 1-i is

12+(-1)2=1+1=2

which gives

(1-i)=2(12-12i)=2(cos蟺4-isin蟺4)

02

Step 2. Using De Moivre's Theorem.

We know that if a complex number is z=r(cos鈭�+isin鈭�)then zn=rn[cos(n鈭�)+isin(n鈭�)].

This gives

role="math" localid="1646763868779" (1-i)5=[2(cos蟺4-isin蟺4)]5=[(2)5(cos{5脳蟺4}-isin{5脳蟺4})]=42(cos5蟺4-isin5蟺4)=42(-12-i(-12))=-4+4i

So,(1-i)5=-4+4i

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