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If z = re^(iθ), what is the value of r if z = 4 + 3i?

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Question: If z = re^(iθ), what is the value of r if z = 4 + 3i?

Options:

  1. 5
  2. 7
  3. 4
  4. 3

Correct Answer: 5

Solution: 

r = |z| = √(4^2 + 3^2) = √(16 + 9) = √25 = 5.

If z = re^(iθ), what is the value of r if z = 4 + 3i?

Practice Questions

Q1
If z = re^(iθ), what is the value of r if z = 4 + 3i?
  1. 5
  2. 7
  3. 4
  4. 3

Questions & Step-by-Step Solutions

If z = re^(iθ), what is the value of r if z = 4 + 3i?
Correct Answer: 5
  • Step 1: Identify the complex number z, which is given as z = 4 + 3i.
  • 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, the real part a = 4 and the imaginary part b = 3.
  • Step 5: Substitute the values into the formula: r = √(4^2 + 3^2).
  • Step 6: Calculate 4^2, which is 16, and 3^2, which is 9.
  • Step 7: Add the results: 16 + 9 = 25.
  • Step 8: Take the square root of 25: √25 = 5.
  • Step 9: Therefore, the value of r is 5.
  • Complex Numbers – Understanding the representation of complex numbers in polar form and calculating their magnitude.
  • Magnitude of Complex Numbers – Using the formula |z| = √(a^2 + b^2) to find the magnitude of a complex number a + bi.
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