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Chapter 9: Q. 51 (page 592)

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

51.(2-i)6

Short Answer

Expert verified

(2-i)6=-23+14.14i

Step by step solution

01

Step 1. Rewrite a+bi as r(cos∅+isin∅).

The magnitude of (2-i)is

role="math" localid="1646806173217" (2)2+(-1)2=2+1=3

This gives

(2-i)=3(23-i13)=3(cos35.264掳+isin35.264掳)

02

Step 2. Using De Moivre's Theorem.

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="1646806473235" (2-i)6=[3(cos35.264掳)-isin(35.264掳)]6={(3)6[cos(6脳35.264)-isin(6脳35.264)]}=27(cos211.584掳-isin211.584掳)=27(-0.8518-i(-0.8518))=-23+14.14i

So,(2-i)6=-23+14.14i

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Most popular questions from this chapter

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