Q. 25 Write each expression in the sta... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 25 (page 630)

Write each expression in the standard form:
${[3 + 4 i]}^{4}$

Short Answer

Expert verified

$- 527 - 336 i$

Step by step solution

Step 1. Given information

Given expression : ${[3 + 4 i]}^{4}$

We have to write it in the standard form.

Step 2. Convert the given expression in the polar form

To solve a complex number raised to (n) it is necessary that it be written in polar form, therefore for the given number we have to :

$z^{4} = {[3 + 4 i]}^{4}$

We know that a complex number can be written in standard form as a + bi now to convert this number into polar f贸rm

$z = (r \cos 胃 + i \sin 胃)$

$\begin{array}{r} r = \sqrt{x^{2} + y^{2}} \\ x = r \cos 鈦� 胃 \\ y = r \sin 鈦� 胃 \end{array}$

Step 3. Calculate r

$\begin{array}{r} r = \sqrt{3^{2} + (4)^{2}} \\ = \sqrt{9 + 16} \\ = \sqrt{25} \\ = 5 \end{array}$

Step 4. Find the angle

$\begin{aligned} x = r \cos 鈦� 胃 \\ \cos 鈦� 胃 = \frac{x}{r} \\ 胃 = \cos^{鈭� 1} 鈦� \frac{x}{r} \\ 胃 = \cos^{鈭� 1} 鈦� \frac{3}{5} \\ = {53.13}^{鈭�} \end{aligned}$

It can be rewritten as:

$z = {[5 (\cos 鈦� {53.13}^{鈭�} + i \sin 鈦� {53.13}^{鈭�})]}^{4}$

Step 5. Write in standard form

To convert complex number raised to the power n,in the standard form, we know that:

$z^{n} = r^{n} (\cos 胃 + i \sin 胃)$

$\begin{aligned} z^{4} = {[5 (\cos 鈦� {53.13}^{鈭�} 鈭� i \sin 鈦� {53.13}^{鈭�})]}^{4} \\ = 5^{4} [\cos 鈦� (4 鈰� {53.13}^{鈭�}) + i \sin 鈦� (4 鈰� {53.13}^{鈭�})] \\ = 625 (\cos 鈦� 212.52 + i \sin 鈦� 212.52) \\ = 625 (鈭� 0.84 鈭� i 0.537) \\ = 鈭� 527 鈭� 336 i \end{aligned}$

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91影视

Write each expression in the standard form:
${[3 + 4 i]}^{4}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Convert the given expression in the polar form

Step 3. Calculate r

Step 4. Find the angle

Step 5. Write in standard form

One App. One Place for Learning.

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91影视

Write each expression in the standard form:3+4i4

Short Answer

Step by step solution

Step 1. Given information

Step 2. Convert the given expression in the polar form

Step 3. Calculate r

Step 4. Find the angle

Step 5. Write in standard form

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Write each expression in the standard form:
${[3 + 4 i]}^{4}$