Question

If \( z = \frac{(\sqrt{3} + i)^3 (3i + 4)^2{(8 + 6i)^2} \), then \( |z| \) is equal to:}

• A. \( 8 \)
• B. \( 2 \)
• C. \( 5 \)
• D. \( 4 \)
• E. \( 10 \)

Hint:

Never expand powers of complex numbers if you only need the modulus. Calculating the modulus of the base and then raising it to the power is much faster.

Answer 1

The Correct Option is B

Concept: The modulus of a complex number obeys several properties that simplify calculations:
• \( |z_1 z_2| = |z_1| \cdot |z_2| \)
• \( |z_1 / z_2| = |z_1| / |z_2| \)
• \( |z^n| = |z|^n \) Instead of simplifying the complex fraction, we calculate the modulus of each component.

Step 1: Calculate the modulus of individual parts.
1. For \( (\sqrt{3} + i) \): \( |\sqrt{3} + i| = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = 2 \). 2. For \( (3i + 4) \): \( |4 + 3i| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5 \). 3. For \( (8 + 6i) \): \( |8 + 6i| = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = 10 \).

Step 2: Apply the modulus properties.
\[ |z| = \frac{|(\sqrt{3} + i)|^3 \cdot |(4 + 3i)|^2}{|(8 + 6i)|^2} \] \[ |z| = \frac{2^3 \cdot 5^2}{10^2} \]

Step 3: Simplify the expression.
\[ |z| = \frac{8 \cdot 25}{100} \] \[ |z| = \frac{200}{100} = 2 \]