Can you please go step by step when solving this? For Exercises 89-92, simplify and write the solution in rectangular form, a + bi. (Hint: convert thecomplex numbers to polar form before simplifying.)(5V2 – 5VZ1)92.(- V3 – i}}

Answered: (5V2 – 5VZ1) 92. (- V3 – i}

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter1: Introduction And Vectors

Section: Chapter Questions

Problem 70P

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Can you please go step by step when solving this?

For Exercises 89-92, simplify and write the solution in rectangular form, a + bi. (Hint: convert the
complex numbers to polar form before simplifying.)
(5V2 – 5VZ1)
92.
(- V3 – i}}

Expert Solution

Step 1: Solution

$\begin{array}{rcl} L e t I & = & \frac{{(5 \sqrt{2} - 5 \sqrt{2} i)}^{4}}{{(- \sqrt{3} - i)}^{3}} = \frac{N}{D} \\ T a l k i n g a b o u t t h e n u m e r a t o r " N " f i r s t, \\ c o n v e r t i n g i t i n t h e p o l a r f o r m : \\ L e t a + b i & = & 5 \sqrt{2} - 5 \sqrt{2} i \\ H e r e, a & = & 5, b = - 5 \sqrt{2} \\ H e n c e, r & = & m a g n i t u d e = \sqrt{a^{2} + b^{2}} \\ = & \sqrt{{(5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{2}} \\ = & \sqrt{100} \\ r & = & 10 \\ N o w, \tan (θ) & = & \frac{b}{a} = \frac{- 5 \sqrt{2}}{5 \sqrt{2}} = - 1 \\ θ & = & - 45 ° \\ N o w, B y D e m o i v r e s t h e o r e m, \\ (r {(\cos (θ) + i \sin (θ))}^{n} & = & r^{n} (\cos (n θ) + i \sin (n θ)) \\ F o r n u m e r a t o r, \\ (10 {(\cos (- 45) + i \sin (- 45))}^{4} & = & 10^{4} (\cos (- 4 \times 45) + i \sin (- 4 \times 45)) \\ = & 10^{4} (- 1 + 0 i) \\ N & = & - 10000 + 0 i . . . . . . . (1) \end{array}$

$\begin{array}{rcl} N o w, T a l k i n g a b o u t t h e d e n o m i n a t o r " D ", \\ c o n v e r t i n g i t i n t h e p o l a r f o r m : \\ L e t a + b i & = & - \sqrt{3} - i \\ H e r e, a & = & - \sqrt{3}, b = - 1 \\ H e n c e, r & = & m a g n i t u d e = \sqrt{a^{2} + b^{2}} \\ = & \sqrt{{(- \sqrt{3})}^{2} + {(- 1)}^{2}} \\ = & \sqrt{4} \\ r & = & 2 \\ N o w, \tan (θ) & = & \frac{b}{a} = \frac{- 1}{- \sqrt{3}} = \frac{1}{\sqrt{3}} \\ θ & = & - 30 ° \\ N o w, B y D e m o i v r e s t h e o r e m, \\ (r {(\cos (θ) + i \sin (θ))}^{n} & = & r^{n} (\cos (n θ) + i \sin (n θ)) \\ F o r n u m e r a t o r, \\ (2 {(\cos (30) + i \sin (30))}^{3} & = & 2^{3} (\cos (3 \times 30) + i \sin (3 \times 30)) \\ = & 8 (0 + i) \\ D & = & 0 + 8 i . . . . . . . (2) \end{array}$

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