How to calculate the power of a complex number?

To calculate the power of a complex number, raise the modulus to the power and multiply the argument by the power.

To calculate the power of a complex number, we use the polar form of the complex number. Let z = r(cosθ + i sinθ) be a complex number in polar form, where r is the modulus and θ is the argument. To find the power of z, we raise the modulus to the power and multiply the argument by the power. That is, z^n = r^n(cos(nθ) + i sin(nθ)).

For example, let z = 2(cosπ/4 + i sinπ/4) be a complex number in polar form. To find z^3, we raise the modulus 2 to the power 3 and multiply the argument π/4 by 3. Thus, z^3 = 8(cos3π/4 + i sin3π/4) = -2√2 - 2√2i.

If the complex number is in rectangular form, we can convert it to polar form using the formulas r = |z| = √(x^2 + y^2) and θ = arg(z) = tan^-1(y/x). Then we can use the formula for the power of a complex number in polar form.

For example, let z = 1 + i be a complex number in rectangular form. We can convert it to polar form as follows: r = |z| = √(1^2 + 1^2) = √2 and θ = arg(z) = tan^-1(1/1) = π/4. Thus, z = √2(cosπ/4 + i sinπ/4). To find z^4, we raise the modulus √2 to the power 4 and multiply the argument π/4 by 4. Thus, z^4 = 4(cosπ + i sinπ) = -4.