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Reviewed CSCA Mathematics question · Hard

Let z = 3 - 4i and w = -2 + i. What is the modulus of z * w?

  1. 5√5
  2. 25
  3. 5
  4. √5
Show the answer and explanation

Correct answer

A. 5√5

Principle or equation

The modulus of a product of complex numbers equals the product of their moduli: |z * w| = |z| * |w|. For a complex number a + bi, the modulus is sqrt(a^2 + b^2).

Why this answer is correct

Compute |z| = sqrt(3^2 + (-4)^2) = sqrt(9 + 16) = 5. Compute |w| = sqrt((-2)^2 + 1^2) = sqrt(4 + 1) = √5. Therefore |z * w| = 5 * √5 = 5√5.

Example

If z = 1 + i and w = 1 - i, then |z| = √2, |w| = √2, so |zw| = 2.

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