Find all roots of the equation z^5=i, i.e. find the five values of i^1/5 and show...

Find all roots of the equation z^5=i, i.e. find the five values of i^1/5 and show them on an Argand diagram. Show all working

Solutions

Expert Solution

zⁿ = rⁿ(cos n + i sin n), where n = power, r = modulus, = angle of radius.

Here n = 1/5, r = 1, ? = /2, z=i^1/5

i^1/5 = [ 1^1/5 (cos + i sin)^1/5 ]

i^1/5 {(cis /2)^1/5 , (cis 3/2)^1/5 , (cis 5/2)^1/5 , (cis 7/2)^1/5 ,(cis 9/2)^1/5 }

i^1/5 {(cis /10), (cis 3/10), (cis 5/10), (cis 7/10), (cis 9/10) }

Note: all of these values are irrational. Also, symbol "cis ": "cis" is a shorthand for cos + i sin

Argand diagram:

milcah answered 1 year ago

Next > < Previous

Related Solutions

Roots of a complex number Find all the solutions to the equation (a) z^ 4 −...

Roots of a complex number Find all the solutions to the equation (a) z^ 4 − 1 = 0 (b) z ^5 + 2^5 i = 0 (c) z^ 6 − 16z^ 3 + 128 = 0

Find all distinct roots (real or complex) of z2+(−6+i)z+(25+15i). Enter the roots as a comma-separated list...

Find all distinct roots (real or complex) of z2+(−6+i)z+(25+15i). Enter the roots as a comma-separated list of values of the form a+bi. Use the square root symbol '√' where needed to give an exact value for your answer. z = ???

Find all the six roots of z = 16−16 sqrt (3i).

1. The equation 8?2 − 3?? = 0 has three roots. Find: (i) The first root,...

1. The equation 8?2 − 3?? = 0 has three roots. Find: (i) The first root, which is between -1 and 0, by the bisection method; (ii) The second root, which is between 1 and 2, by Newton’s method; (iii) The third root, which is between 3 and 4, by both the bisection method and Newton’s method. All the answers should be correct to 2 decimal places.

Let z = 4 - 4i. find the sixth roots of z and represent them in...

Let z = 4 - 4i. find the sixth roots of z and represent them in the plane.

Find all primitive roots: (a) modulo 25, or show that there are none (b) modulo 34,...

Find all primitive roots: (a) modulo 25, or show that there are none (b) modulo 34, or show that there are none (c) Assuming that 2 is a primitive root modulo 67, find all primitive roots modulo 67.

Use Newton's method to find all real roots of the equation correct to six decimal places....

Use Newton's method to find all real roots of the equation correct to six decimal places. (Enter your answers as a comma-separated list.) 8/x = 1 + x^3

Find the following z values for the standard normal variable Z. Use Table 1. (Negative values...

Find the following z values for the standard normal variable Z. Use Table 1. (Negative values should be indicated by a minus sign. Round your answers to 2 decimal places.) a. P(Z ≤ z) = 0.1105 b. P(z ≤ Z ≤ 0) = 0.1607 c. P(Z > z ) = 0.7698 d. P(0.25 ≤ Z ≤ z) = 0.3428

What is the Regression Equation (i.e., what are the values of a and b)?

What is the Regression Equation (i.e., what are the values of a and b)? What is the forecast value for November? What is the Standard Error of the Estimate?

Find all solutions of the equation x^{5}=2

Subjects