Question

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

• A. \(\frac{2}{25}\)
• B. \(-\frac{2}{25}\)
• C. \(\frac{1}{25}\)
• D. \(-\frac{1}{25}\)

Hint:

For modulus, use \(|z^n|=|z|^n\) and \(\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks for the modulus of a complex number that is a result of division and exponentiation. We need to use the properties of modulus.
Step 2: Key Formula or Approach:
The key properties of the modulus are: 1. $|z^n| = |z|^n$ 2. $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$ 3. $|a+bi| = \sqrt{a^2+b^2}$
We will apply these properties to find the modulus without fully expanding the complex number.
Step 3: Detailed Explanation:
Let $z = \frac{(1+i)^{10}}{(2i-4)^4}$. We want to find $|z|$. Using the properties of modulus: \[ |z| = \left|\frac{(1+i)^{10}}{(-4+2i)^4}\right| = \frac{|(1+i)^{10}|}{|(-4+2i)^4|} = \frac{|1+i|^{10}}{|-4+2i|^4} \] First, calculate the modulus of the base of the numerator: \[ |1+i| = \sqrt{1^2 + 1^2} = \sqrt{2} \] So, the modulus of the numerator is $|1+i|^{10} = (\sqrt{2})^{10} = 2^{10/2} = 2^5 = 32$.
Next, calculate the modulus of the base of the denominator: \[ |-4+2i| = \sqrt{(-4)^2 + 2^2} = \sqrt{16+4} = \sqrt{20} \] So, the modulus of the denominator is $|-4+2i|^4 = (\sqrt{20})^4 = (20^{1/2})^4 = 20^2 = 400$.
Finally, divide the modulus of the numerator by the modulus of the denominator: \[ |z| = \frac{32}{400} \] Simplify the fraction: \[ |z| = \frac{16}{200} = \frac{8}{100} = \frac{2}{25} \] Step 4: Final Answer:
The modulus of the complex number is $\frac{2}{25}$. The modulus is always a non-negative real number, so options (B) and (D) are incorrect by definition. Therefore, option (A) is correct.