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

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

Options:

  1. 4√2
  2. 8
  3. 4
  4. 2√2

Correct Answer: 4√2

Solution: 

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

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

Practice Questions

Q1
If z = re^(iθ), what is the value of r if z = 4 + 4i?
  1. 4√2
  2. 8
  3. 4
  4. 2√2

Questions & Step-by-Step Solutions

If z = re^(iθ), what is the value of r if z = 4 + 4i?
Correct Answer: 4√2
  • 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.
  • Complex Numbers – Understanding the representation of complex numbers in polar form and calculating their magnitude.
  • Magnitude Calculation – Using the formula for the magnitude of a complex number, |z| = √(a^2 + b^2), where z = a + bi.
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