Final answer:
The equation z⁴ = -625 has a solution with an argument between 270° and 360°, which is 5cis(315°) or approximately 3.535 + i(-3.535) when converted to Cartesian form and rounded to three decimal places.
Step-by-step explanation:
To find the solution for the equation z⁴ = -625 with an argument between 270° and 360°, we first need to recognize that -625 can be rewritten as 625 times -1, or in polar form as 625cis(180°) since the argument of -1 is 180°.
Since the fourth power root is taken, the resulting arguments will be equally spaced in a full circle.
This leads to four roots, each having arguments that are 45° apart, starting from 180°/4 = 45°.
The root that is strictly between 270° and 360° is obtained by adding 270° to the initial 45° argument of the first root.
The third root, therefore, will have an argument of 45° + 2 * 90° = 225°, and the fourth root will have an argument of 45° + 3 * 90° = 315°, which is indeed strictly between 270° and 360°.
The magnitude of each root is the fourth root of 625, which is √(625) = 5, since 5⁴ = 625.
Hence, the complex number z that satisfies the conditions is 5cis(315°) or in Cartesian form, z = 5(cos(315°) + isin(315°)).
This simplifies to z ≈ 3.535 + i(-3.535), rounded to three decimal places as we are using the approximate values of cos(315°) and sin(315°).
Hence, a solution with an argument between 270° and 360°, which is 5cis(315°) or approximately 3.535 + i(-3.535)