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If z = 1 + i√3, what is the value of z^2? (2023)

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Question: If z = 1 + i√3, what is the value of z^2? (2023)

Options:

  1. -2 + 2i√3
  2. 4
  3. 1 - 2i√3
  4. 1 + 2i√3

Correct Answer: -2 + 2i√3

Exam Year: 2023

Solution: 

z^2 = (1 + i√3)^2 = 1^2 + 2(1)(i√3) + (i√3)^2 = 1 + 2i√3 - 3 = -2 + 2i√3.

If z = 1 + i√3, what is the value of z^2? (2023)

Practice Questions

Q1
If z = 1 + i√3, what is the value of z^2? (2023)
  1. -2 + 2i√3
  2. 4
  3. 1 - 2i√3
  4. 1 + 2i√3

Questions & Step-by-Step Solutions

If z = 1 + i√3, what is the value of z^2? (2023)
  • Step 1: Start with the given value of z, which is z = 1 + i√3.
  • Step 2: To find z^2, we need to calculate (1 + i√3)^2.
  • Step 3: Use the formula for squaring a binomial: (a + b)^2 = a^2 + 2ab + b^2.
  • Step 4: Here, a = 1 and b = i√3. So, we calculate each part:
  • Step 5: Calculate a^2: 1^2 = 1.
  • Step 6: Calculate 2ab: 2(1)(i√3) = 2i√3.
  • Step 7: Calculate b^2: (i√3)^2 = i^2 * (√3)^2 = -1 * 3 = -3.
  • Step 8: Now combine all parts: z^2 = 1 + 2i√3 - 3.
  • Step 9: Simplify the expression: 1 - 3 + 2i√3 = -2 + 2i√3.
  • Step 10: Therefore, the value of z^2 is -2 + 2i√3.
  • Complex Numbers – Understanding the properties and operations involving complex numbers, including squaring a complex number.
  • Algebraic Expansion – Applying the binomial theorem or algebraic identities to expand expressions.
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