Question

The modulus of the complex number \[ \frac{(1 + i)^{10} (2 - i)^6}{(2i - 4)^4} \] is equal to:

• A. 8
• B. 10
• C. 16
• D. 30
• E.

Hint:

To calculate the modulus of a complex number raised to a power, first find the modulus of the number, raise it to the power, and then apply the modulus to the entire expression. Use the property that \( |a \cdot b| = |a| \cdot |b| \).

Answer 1

The Correct Option is B

Let’s first find the modulus of each complex number. The modulus of a complex number \( z = a + bi \) is given by: \[ |z| = \sqrt{a^2 + b^2} \] Step 1: Calculate the modulus of each part.
1. Modulus of \( (1 + i) \): \[ |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2} \] So, \( |(1 + i)^{10}| = (\sqrt{2})^{10} = 2^5 = 32 \). 2. Modulus of \( (2 - i) \): \[ |2 - i| = \sqrt{2^2 + (-1)^2} = \sqrt{5} \] So, \( |(2 - i)^6| = (\sqrt{5})^6 = 5^3 = 125 \). 3. Modulus of \( (2i - 4) \): \[ |2i - 4| = \sqrt{(-4)^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \] So, \( |(2i - 4)^4| = (2\sqrt{5})^4 = 4^2 \cdot 5^2 = 16 \cdot 25 = 400 \).
Step 2: Now, calculate the modulus of the entire expression: \[ \left| \frac{(1 + i)^{10} (2 - i)^6}{(2i - 4)^4} \right| = \frac{|(1 + i)^{10}| \cdot |(2 - i)^6|}{|(2i - 4)^4|} \] Substitute the values we calculated: \[ = \frac{32 \cdot 125}{400} = \frac{4000}{400} = 10 \] Thus, the correct answer is option (B), 10.