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

Q: Find all complex numbers z satisfying 4 = -81

A: Find all the complex numbers z satisfying z4= -81

Q: 7. The angle in evaluating the complex number in transcendental function should be expressed in…

A: Here, we shall check each problem to conclude whether it is true or false. 7. The angle in…

Q: 4. Find the modulus, Iz], and argument, 0 , of the complex number. -1 + i

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Q: If a complex number z satisfies z = z-1, what can be said about the number? Are con Nay/ Say that…

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Q: What we want to do next is to write a procedure to raise any complex number to any integer power (x…

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Q: What happens when a complex number is multiplied by its conjugate? It turns into a real number…

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Q: Find the modulus and the principal value of the argument of the following powers of complex number:…

A: Explanation of the answer is as follows

Q: 38. Let a be a complex number that is algebraic over Q and let r be a rational number. Show that a'…

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Q: Suppose z is a complex number that possesses a fourth root w that is neither real nor pure…

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

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Q: φι dz (z-zo)" [2πί το n = 1 n>1

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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: Find the roots of complex numbers √√64 e¹.4i if n=4, k= 0,1,2,3 k=0, z1= k=1,z2= k=2, z3= k=3, z4=

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

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Q: Prove:The addition of complex number is associative.

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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: Find the roots of the given complex number 1. z^3 = -2

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

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Q: Consider the complex number z = 10250°. What is the smallest positive integer n such that z" is a…

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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: 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 a be a complex number that is algebraic over Q and let r be arational number. Show that ar is…

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Q: p and q are complex numbers such that pl = 7√3 and lp + ql = 12√2. - On what interval must q fall…

A: Use  Cauchy inequalities and solve the question.

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: Let C be a fixed nonzero complex number and note that i is a multiple valued, what restriction must…

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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: Find the modulus (r) for the given complex number z = -6.3 + 8.2i.

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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: 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: Give an example of a reduced cubic in the form x = 3px + 2q with three real roots but where q – p³ <…

A: Question: Give an example of a reduced cubic in the form x3=3px+2q with three real roots but where…

Q: For a positive integer n, the complex number z = r(cos 0 + i sin 0) has exactly n distinct nth roots…

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

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Q: 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…

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Q: Let z be a complex number. Then the solution(s) of sinhz+coshz-e is (are)* None of these z=nti+1…

A: See solution below

Q: Use Bombelli’s approach to find a complex number c = a + bi such that c3= 18 + 26i where a and b are…

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Q: Let z = 3 – Ai, w = 1+ 2i be complex numbers. For both z and w, what is the: • Real part • Imaginary…

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Q: 1. Find all complex 4th roots of -8 – 8/3i, plot their location in the complex plane and show that…

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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: 8. The number of roots of a complex is equal to the index of the radical sign True O False

A: The radical operation can also be performed for a given complex number. The nth radical of a complex…

Q: Let z be a non-zero complex number. D if OsArg(z)sTt/2 D for all complex numbers D if OsArg(z)sn…

A: we know that logzn≠nlogz for all complex numbers. Since ew=z and -π&lt;Imw≤π If w=logz, nw is a…

Q: Let Re(z) and Im(z) be the real and imaginary parts respectively of a complex number z . If there…

A: Let z=a+ib   and     w=u+iv.

Q: p and q are complex numbers such that |a| = 4√5 and lp + ql = 9√/2. - On what interval must lp fall…

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Q: The division of a complex number z, by anon- zero complex number z, is defined as the multiplication…

A: Let, Z1= a1+ib1 and, Z2=a2+ib2

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: 2+31 In(2+) . Evaluate the complex number 3441

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