Question

Consider z=3√(3)+3i. What happens to the modulus and argument when z is raised to the 4th power?

A. The modulus increases by a factor of 216, and the argument increases by π/2
B. The modulus increases by a factor of 216, and the argument increases by 2π/3
C. The modulus increases by a factor of 1,296, and the argument increases by π/2
D. The modulus increases by a factor of 1,296, and the argument increases by 2π /3

Answer 1

The modulus of z=3√(3)+3i when raised to the 4th power increases by a factor of 1,296 and the argument increases by 2π/3, making the correct answer to the question D.

Step-by-step explanation:

When raising a complex number to a power, the modulus (absolute value) of the number is raised to that power, and the argument (angle) is multiplied by that power. In the given case of z = 3√(3) + 3i, we find the modulus and argument before raising to the power.

The modulus of z is √((3√(3))² + (3)²), which simplifies to 3√(4) = 3√(2²) = 6. The argument can be found using arctan of the imaginary part over the real part, which is arctan(1/√(3)) = π/6. If we raise z to the 4th power, the modulus becomes 6⁴ = 1296, and the argument becomes 4(π/6) = 2π/3.

Thus, the correct answer is D. The modulus increases by a factor of 1,296, and the argument increases by 2π /3.