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:

When asked for the modulus of a complex expression involving products, divisions, or powers, never try to simplify the complex number first. It's much faster and easier to use the properties of modulus (\(|z_1 z_2| = |z_1||z_2|\), \(|z_1/z_2| = |z_1|/|z_2|\), \(|z^n| = |z|^n\)) to calculate the modulus of each part separately and then combine them.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to find the modulus of a complex number which is given in a fractional form with powers.

Step 2: Key Formula or Approach:
We will use the properties of modulus:
1. \(|z_1/z_2| = |z_1|/|z_2|\)
2. \(|z^n| = |z|^n\)
3. The modulus of a complex number \(z = a+bi\) is \(|z| = \sqrt{a^2+b^2}\).
Using these properties, we can find the modulus of the numerator and denominator separately without actually computing the complex number itself.

Step 3: Detailed Explanation:
Let \(z = \frac{(1+i)^{10}}{(2i-4)^4}\). We need to find \(|z|\).
Using the properties of modulus:
\[ |z| = \left| \frac{(1+i)^{10}}{(-4+2i)^4} \right| = \frac{|(1+i)^{10}|}{|(-4+2i)^4|} = \frac{|1+i|^{10}}{|-4+2i|^4} \]

Step 3a: Calculate the modulus of the numerator term
First, find the modulus of \(1+i\):
\[ |1+i| = \sqrt{1^2 + 1^2} = \sqrt{2} \]
Now, raise it to the power of 10:
\[ |1+i|^{10} = (\sqrt{2})^{10} = (2^{1/2})^{10} = 2^5 = 32 \]

Step 3b: Calculate the modulus of the denominator term
First, find the modulus of \(-4+2i\):
\[ |-4+2i| = \sqrt{(-4)^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} \]
We can simplify \(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\).
Now, raise it to the power of 4:
\[ |-4+2i|^4 = (\sqrt{20})^4 = (20^{1/2})^4 = 20^2 = 400 \]
Alternatively, \((2\sqrt{5})^4 = 2^4 \times (\sqrt{5})^4 = 16 \times 5^2 = 16 \times 25 = 400\).

Step 3c: Combine the results
\[ |z| = \frac{32}{400} \]
Simplify the fraction:
\[ |z| = \frac{16}{200} = \frac{8}{100} = \frac{2}{25} \]

Step 4: Final Answer:
The modulus of the complex number is \(\frac{2}{25}\). Note that modulus must be a non-negative real number, so options (B) and (D) are incorrect by definition.