12 3. F(s) = = 53 +8 + sin 2t - 1). Suggestion: After finding the three cube roots -2 and 1±√√√3i of -8, it is helpful to notice that the property z + z = 2 Re z of complex numbers enables one to write ei√√3t e-i√√31 pi√31 + -1+i√√√3 = 2 Re -1-i√√√3 -1+i√√√3 Ans. f(t) = e−2+ + et (√√3 sin √√3t - cos√3t). +e(√√3

Q: 1. Evaluate in rational form (no decimals). (4)(3°) a) (32°*)(32') (4)(3²) b) 32

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Q: 4. (choose one.) (a) Find all 3rd roots of –8i. (b) Find all complex roots of the equation 26 = –9

A: 4) a) Given:          cube roots of -8i we write in trigonometric form:          =8cos-π2+isin-π2 we…

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A: This is a problem of complex analysis.

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A: see next step

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A: According to guidelines i solve first question only so kindly request you to repost the remaining…

Q: 30. The complex number 1 + V3i can be expressed in exponential form as 2e A в 2e- is 2e- 2e D

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A: The complex number z is defined as z=i1+i1-2i2

Q: a) w=−√3−3i    evaluate (−√3−3i)^15 (that would mean writing w^15 in the a+bi form)

A: Express in polar form, as follows:

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A: on based data given detailed solution provided

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Q: Number 8

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Q: Find the 5th roots of 9. -16/2 + 16vZi

A: =12+32i9De Moivre's theorem;=cosπ3+i sinπ39= eiπ39=ei3π=cos3π + isin3π=-1

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A: From the question image that you have uploaded, only 1 complete question is visible, which is the…

Q: (d) Let Z₁ = −4+ 8i and Z₂ = -1 -2i. Show that (i) 21/2₂ | = 4. | (ii) (iii) |z₁|² +Z₁ Z₂ = 100.…

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Q: #7 using Descartes's rule of sign

A: Part (a) The polynomial f(x) = x3 +3x +7 has a degree 3, so, it has 3 roots. For positive real…

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A: The solution is :

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Q: Express each one of the following in the standard form a + ib. 1 (1+ i)? (iii) 3 - i 1 (vi) 1- 2i…

A: vii)

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Q: Use the defin a) 7+√√=9 2+3i a = 2 b=3 b) √-16 पा I ▬▬▬▬▬▬▬▬ =4 b =

A: a) 7+sqrt(-9)b) sqrt(-16)c) 3

Q: 5. a. Given z, = -7 + 6i(15+2) and z, = -2i(36-0) + 8i , find |z, + z2]. 4 b. Simplify (+ ), and…

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