3. An nth root of unity is a complex number z such that z" = 1. For a given n, how many n'h roots of unity are there and where in the complex plane are they located? Justify your answers.

Q: Let z be a complex number. Then sin(3z)-i is bounded in C. * True False

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Q: Let z be a complex number Then the solution(s) of sinhzicoshz31/e is (are) None of these. does not…

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Q: Find the 5™ roots of the complex number [(2+x) – 5i] , where x = last digit of your ID Number.

A: To find- Find the 5th roots of the complex number 2 + x - 5i, where x = 2

Q: Prove the Euler's formula: e" = cos0+ jsin 0 using the definition of the complex number

A: We have to prove that ejθ=cosθ+jsinθ. We know that ex=1+x+x22!+x33!+x44!+.......... Now,…

Q: 1. ________ Theorem states that if z = r(cos θ + i sin θ) is a complex number and n is a positive…

A: Given: 1. ________ Theorem states that if z=rcos θ+i sin θ is a complex number and n is a positive…

Q: 5) Compute the fllowing complex numbers in standard arbi form for a, bE R. List all values if there…

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Q: express  in the form z = x +yi the complex number z whose modulus and argument are given.  |z| =…

A: z= x+iyz=12,    arg(2) =-π3modulus of z =z = x2+y2 =12⇒x2+y2 =12     -(i)

Q: 5. Determine the roots of the complex numbers studied in class. z³ = 2 – 3i

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Q: How may the following complex numbers be obtained fromz = x + iy geometrically? Sketch. 1 / z

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Q: Given that the complex numbers zı and z2 are such that z, = cos 10° + i sin 10° and z2 cos 15° + i…

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Q: 2) Suppose 1 and w are complex numbers, show that Aw =

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Q: iv) in complex number z3. (7)? = |z|?. z

A: Complex number: A complex number has the form z=x+ιy, where x is called the real part and y is the…

Q: Determine the roots of the complex numbers discussed in class. 1. z3 = -2 2. z3 = -3

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Q: Every complex number a has a 4-bth root in C That is, there exists a complex Mumber w such that…

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Q: 4. If the two complex numbers z1=ri(cos01+ i sin 01) and z2= r2(cos 02+ i sin 02) then the numbers…

A: The given complex numbers,z1=r1cosθ1+i sinθ1 and z2=r2cosθ2+i sinθ2

Q: Take the last four digits ABCD of your student identification number, and let z1= A + Bi and z2 = C…

A: Here, z1=5i z2=6+4i

Q: Let z be a complex number. Then the solution(s) of sinhz+coshz=1/e is (are) z=2nti-1 where n is an…

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Q: Find the cube roots of the complex number z = 125 (cos(60°) + i sin(60°)). Select all correct…

A: We have to find cube root of given complex number.

Q: Find all complex numbers satisfying |z + 3 + 4i| = |z − 5|

A: Topic:- complex numbers

Q: 4. Use DeMoivre's Theorem to find the power of the complex number [4 (cos 50 ° + i sin 50°) ]3

A: Use demovirs theorem

Q: Find all the nth roots of z. Express answers in polar form z = 4 + 4i √3 , n = 3

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Q: 5. Determine the roots of the complex numbers studied in class. z3 = 2 – 3i %3|

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Q: How may the following complex numbers be obtained fromz = x + iy geometrically? Sketch. -z

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Q: Let z be a complex number. Then sinh(2z)-i is bounded in C.* False True

A: Given a statement

Q: 1. If z = 1+ iy is a complex number. Use the exponential form to compute (1+ /3 i)201

A: Definition used -            DeMoivre's theorem -             For a complex number z=r(cosθ+isinθ)…

Q: V3 1 (- 2

A: given(-32-12i)20 =(-1)20(32+12i)20 =(cos30+isin30)20 The formula of Demoivre's theorem is cos θ+isin…

Q: Find all the complex fourth roots in rectangular form of w = 64 cos +i sin %3D (Type your answer in…

A: To find the fourth root of the given complex number.

Q: Let z be a complex number. Then the solution(s) of sinhz+coshz=1/e is (are) * None of these does not…

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Q: 4. This question has two parts. a) Compute the value of eln 2020+3xi/4 b) Find all complex numbers z…

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Q: Determine the roots of the complex numbers studied in class. 28. z5 = 4i 29. z5 = 6i

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Q: Photo #1: Question Photo #2: Answer Key

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Q: - Find the roots of the following complex nos. 3. %3D 3. Z =\ Z-27-0 %3D 3. |

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Q: 5. Find the 3 3rd roots of -27i, and graph them in the complex plane.

A: -27i=27cos-π2+isin-π2 Now, use De Moivre's Formula

Q: Find the roots of the given complex number 1. z^5 = 4i

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Q: The modulus of a complex number x+jy is equal to

A: The modulus of a complex number x+jy is equal to

Q: Consider the following complex number. Z = 4(cos (280°) + i sin (280°)) (a) Find two square roots…

A: Root of complex number

Q: Let z be a complex number. Then the solution(s) of sinhz2+coshz=1/e is (are) * O z=2nni-1 where n is…

A: Solution:

Q: Let z be a complex number. Then the solution(s) of sinhz+coshz=1/e is (are) * O z=2nti-1 where n is…

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Q: Let z be a complex number satisfying 12|z|² = 2|z + 2|² + |z² + 1|² + 31. What is the value of z +-?…

A: Given that Let z be a complex number satisfying  12z2=2z+22+z2+12+31 To find the value of z+6z

Q: A (real or complex) scalar u is said to be an nth rootof unity if un = 1. Let ωn = e 2πi n . Use…

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Q: 5. Find and sketch on the complex plane the 7 seventh roots of 128. Repeat the operation for 128j.…

A: We have to find and sketch on the complex plane the seventh roots of 12.

Q: Let z be a complex number .then e^(3z+i) is bounded over the set of complex numbers?

A: Let z is a complex number. Let f(z) = e(3z+i)  Let us assume that f(z) is bounded. Hence f(z) is an…

Q: z = r(cos 0+isin 0).

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Q: Fihd the cube roots of the complex ber z = 125 (cos(210°) + i sin(210°)). Select all correct…

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Q: Let z be a complex number. Then cos(72)+3i is unbounded in C.

A: A complex no. in the form of a+ ib

Q: a) Express the complex number Z 10 (cos 150° +i sin 150° ) in rectangular and exponential form. V3…

A: (a)z=10(cos 150°+i sin 150)where r=10and θ=150*π180=56πExponential form is z=reiθz=10ei5π/6In…

Q: Q2\ If z = 1 + i, what is z + z? + z + 1 as a single complex number?

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Q: Let z be a complex number. Then sinh(2z)-i is bounded in C. * False True

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Q: 4 9) Use Euler's formula to compute (--i. Express as a 2 2 complex number of the form a + bi.

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Q: If a is a complex number such th equation az²+z+1=0 has one p

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