If z = re^(iθ), what is the value of r if z = 4 + 4i?

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Q: If z = re^(iθ), what is the value of r if z = 4 + 4i?

Solution: r = |z| = √(4^2 + 4^2) = √(16 + 16) = √32 = 4√2.

Steps: 9

Step 1: Identify the complex number z, which is given as z = 4 + 4i.
Step 2: Recall that in the expression z = re^(iθ), r represents the magnitude (or absolute value) of the complex number z.
Step 3: To find the magnitude r, use the formula r = |z| = √(a^2 + b^2), where a is the real part and b is the imaginary part of z.
Step 4: In our case, a = 4 and b = 4.
Step 5: Substitute the values into the formula: r = √(4^2 + 4^2).
Step 6: Calculate 4^2, which is 16. So, we have r = √(16 + 16).
Step 7: Add the two values: 16 + 16 = 32.
Step 8: Now, calculate the square root: r = √32.
Step 9: Simplify √32: √32 = √(16 * 2) = √16 * √2 = 4√2.