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:

Use \(|z^n|=|z|^n\) and \(\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}\). Modulus is always non-negative.

Answer 1

The Correct Option is A

We need to find the modulus of \[ \frac{(1+i)^{10}}{(2i-4)^4}. \] Using the property of modulus: \[ \left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}. \] So, \[ \left|\frac{(1+i)^{10}}{(2i-4)^4}\right| = \frac{|(1+i)^{10}|}{|(2i-4)^4|}. \] Using \[ |z^n|=|z|^n, \] we get \[ = \frac{|1+i|^{10}}{|2i-4|^4}. \] Now, \[ |1+i|=\sqrt{1^2+1^2}. \] \[ |1+i|=\sqrt{2}. \] Therefore, \[ |1+i|^{10}=(\sqrt{2})^{10}. \] \[ =(2^{1/2})^{10}. \] \[ =2^5=32. \] Now, \[ 2i-4=-4+2i. \] So, \[ |2i-4|=\sqrt{(-4)^2+2^2}. \] \[ =\sqrt{16+4}. \] \[ =\sqrt{20}. \] \[ =2\sqrt{5}. \] Therefore, \[ |2i-4|^4=(2\sqrt{5})^4. \] \[ =(\sqrt{20})^4. \] \[ =20^2=400. \] Hence, \[ \left|\frac{(1+i)^{10}}{(2i-4)^4}\right| = \frac{32}{400}. \] \[ =\frac{2}{25}. \] Therefore, the modulus is \[ \frac{2}{25}. \]