To calculate $z \times w$, we need to use the properties of multiplying complex numbers in polar form. Given:

$z = 1.25 e^{0.61 i}, \quad w = 3.66 e^{-2.31 i}$

The multiplication of two complex numbers in polar form is given by the rule:

$z \times w = r_1 r_2 e^{i(\theta_1 + \theta_2)}$

Where:

• $r_1$ and $r_2$ are the magnitudes of $z$ and $w$, respectively.
• $\theta_1$ and $\theta_2$ are the angles of $z$ and $w$, respectively.

Step 1: Multiply the magnitudes

$r_1 = 1.25, \quad r_2 = 3.66$ $r_1 \times r_2 = 1.25 \times 3.66 = 4.575$

Step 2: Add the angles

$\theta_1 = 0.61, \quad \theta_2 = -2.31$ $\theta_1 + \theta_2 = 0.61 + (-2.31) = -1.70$

Step 3: Write the product

$z \times w = 4.575 e^{-1.70 i}$

This is the product of $z$ and $w$ in polar form. You can also convert it back to rectangular (Cartesian) form if needed.

Would you like further details, such as converting to rectangular form or visualization?

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Here are 5 related questions:

1. How do you convert polar form of complex numbers to rectangular form?
2. What are the properties of multiplying complex numbers in polar form?
3. How can you divide two complex numbers in polar form?
4. What is the physical significance of the argument of a complex number?
5. How can you plot complex numbers in polar and Cartesian coordinates?

Tip: When multiplying complex numbers in polar form, always multiply the magnitudes and add the arguments (angles).