Write the exponential form and the principal argument of the following complex numbers.(i) z =i/(−2 − 2i)

To find the exponential form and the principal argument of the complex number z = i/(−2 − 2i), we first need to simplify the complex number.

Step 1: Simplify the complex number Multiply the numerator and the denominator by the conjugate of the denominator to get rid of the imaginary part in the denominator.

z = i/(−2 − 2i) * (−2 + 2i)/(−2 + 2i) = −2i + 2i² / (4 - 4i²) = −2i - 2 / 4 + 4 = -2 - 2i / 8 = -1/4 - 1/4i

So, the simplified form of the complex number is z = -1/4 - 1/4i.

Step 2: Convert to polar form The polar form of a complex number is r(cos θ + i sin θ), where r is the magnitude of the complex number and θ is the argument of the complex number.

The magnitude r is given by √((real part)² + (imaginary part)²) = √((-1/4)² + (-1/4)²) = √(1/16 + 1/16) = √(1/8) = √2/4 = 1/2.

The argument θ is given by arctan(imaginary part / real part) = arctan(-1/4 / -1/4) = arctan(1) = π/4.

So, the polar form of the complex number is z = 1/2(cos π/4 + i sin π/4).

Step 3: Convert to exponential form The exponential form of a complex number is re^(iθ), where r is the magnitude of the complex number and θ is the argument of the complex number.

So, the exponential form of the complex number is z = 1/2e^(iπ/4).

Therefore, the exponential form of the complex number z = i/(−2 − 2i) is z = 1/2e^(iπ/4), and the principal argument of the complex number is π/4.