Final answer:
To find the polar form of the roots of (-14+3i)⁻²/⁵, calculate the complex number's magnitude and angle, then apply the exponent by dividing the angle by 5 and taking the fifth root of the modulus, considering the rotation implied by the negative power.
Step-by-step explanation:
The roots of the complex number (-14+3i)⁻²/⁵ can be expressed in polar form using the modulus (magnitude) and argument (angle) of the number. First, we need to find the modulus and argument of (-14 + 3i). The modulus of a complex number a + bi is √(a² + b²), and the argument is atan2(b, a), which gives us the angle formed with the positive x-axis in a counterclockwise direction.
After computing the modulus and argument of our complex number, we then can calculate the fifth root by dividing the argument by 5 and taking the fifth root of the modulus, while also taking into account the negative sign which implies a further rotation by π radians or 180 degrees. Finally, we write the result as r(cos θ + i sin θ) where r is the magnitude and θ is the angle in radians.