Chapter 9: Q. 52 (page 592)

In Problems 41–52, write each expression in the standard form $a + b i$ .
$52 . {(1 - \sqrt{5} i)}^{8}$

Short Answer

Expert verified

${(1 - \sqrt{5} i)}^{8} = - 1263.6 - 286.42 i$ .

Step by step solution

Step 1. Rewrite a+bi in the form of r(cos∅+isin∅).

The magnitude of $(1 - i \sqrt{5})$ is

$\sqrt{1^{2} + {(- \sqrt{5})}^{2}} = \sqrt{1 + 5} = \sqrt{6}$

which gives

$(1 - i \sqrt{5}) = \sqrt{6} (\frac{1}{\sqrt{6}} - \frac{\sqrt{5}}{\sqrt{6}} i) = \sqrt{6} (\cos 294.1 ° - i \sin 294.1 °)$

Step 2. Using De Moivre's Theorem.

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

We get

role="math" localid="1646807571178" ${(1 - i \sqrt{5})}^{8} = {[\sqrt{6} (\cos 294.1 ° - i \sin 294.1 °)]}^{8} = {(\sqrt{6})^{8} [\cos (8 \times 294.1) - i \sin (8 \times 294.1)]} = 1296 (\cos 2352.8 ° - i \sin 2352.8 °) = 1296 (- 0.975 - i 0.221) = - 1263.6 - 286.42 i$

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In Problems 41–52, write each expression in the standard form $a + b i$ .
$52 . {(1 - \sqrt{5} i)}^{8}$

Short Answer

Step by step solution

Step 1. Rewrite a+bi in the form of 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.52.(1-5i)8

Short Answer

Step by step solution

Step 1. Rewrite a+bi in the form of 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

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

In Problems 41–52, write each expression in the standard form $a + b i$ .
$52 . {(1 - \sqrt{5} i)}^{8}$