Chapter 9: Q. 50 (page 592)

In Problems 41–52, write each expression in the standard form $a + b i$ .
$50 . {(\sqrt{3} - i)}^{6}$

Short Answer

Expert verified

${(\sqrt{3} - i)}^{6} = - 64$

Step by step solution

Step 1. Rewrite (3-i) in the form r(cos∅+isin∅).

The magnitude of $(\sqrt{3} - i)$ is

$\sqrt{{(\sqrt{3})}^{2} + {(- 1)}^{2}} = \sqrt{3 + 1} = \sqrt{4} = 2$

which gives

$(\sqrt{3} - i) = 2 (\frac{\sqrt{3}}{2} - i \frac{1}{2}) = 2 (\cos 330 ° - i \sin 330 °)$

Step 2. Using De Moivre's Theorem.

If a complex number is $z = r (\cos \emptyset + i \sin \emptyset)$ then $z^{n} = r^{n} [\cos (n \emptyset) + i \sin (n \emptyset)]$ .

This gives

role="math" localid="1646805734516" ${(\sqrt{3} - i)}^{6} = {[2 (\cos 330 ° - i \sin 330 °)]}^{6} = {(2)^{6} [\cos (6 \times 330 °) - i \sin (6 \times 330 °)]} = 64 (\cos 1980 ° - i \sin 1980 °) = 64 (\cos 180 ° - i \sin 180 °) = 64 (- 1 + 0 i) = - 64$

So ${(\sqrt{3} - i)}^{6} = - 64$

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In Problems 41–52, write each expression in the standard form $a + b i$ .
$50 . {(\sqrt{3} - i)}^{6}$

Short Answer

Step by step solution

Step 1. Rewrite (3-i) in the form r(cos∅+isin∅).

Step 2. Using De Moivre's Theorem.

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In Problems 41–52, write each expression in the standard form a+bi.50.(3-i)6

Short Answer

Step by step solution

Step 1. Rewrite (3-i) in the form r(cos∅+isin∅).

Step 2. Using De Moivre's Theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Study anywhere. Anytime. Across all devices.

In Problems 41–52, write each expression in the standard form $a + b i$ .
$50 . {(\sqrt{3} - i)}^{6}$