Questions: Determine the standard polar form of the complex number z=-4+4√3i. Write the argument in terms of radians. Complete parts a through f below. a. What is the value of r ? r= (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Solution Steps

To determine the standard polar form of the complex number \( z = -4 + 4\sqrt{3}i \), we need to find the magnitude \( r \) and the argument \( \theta \). The magnitude \( r \) is given by the formula \( r = \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the real and imaginary parts of the complex number, respectively.

1. Identify the real part \( x \) and the imaginary part \( y \) of the complex number.
2. Use the formula \( r = \sqrt{x^2 + y^2} \) to calculate the magnitude \( r \).

For the complex number \( z = -4 + 4\sqrt{3}i \), we identify the real part \( x \) and the imaginary part \( y \):

• \( x = -4 \)
• \( y = 4\sqrt{3} \)

The magnitude \( r \) is calculated using the formula: \[ r = \sqrt{x^2 + y^2} \] Substituting the values of \( x \) and \( y \): \[ r = \sqrt{(-4)^2 + (4\sqrt{3})^2} = \sqrt{16 + 48} = \sqrt{64} = 8 \]

Final Answer