If $x + iy = \sqrt{\frac{a+ib}{c+id}}$, then what is $(x^2 + y^2)^2$?

Steps to Solve

1. Find the magnitude of $x + iy$

   We are given $x + iy = \sqrt{\frac{a+ib}{c+id}}$. We want to find $x^2 + y^2$, which is related to the magnitude of the complex number $x + iy$. Recall that for a complex number $z = x + iy$, the magnitude $|z|$ is given by $|z| = \sqrt{x^2 + y^2}$, and $|z|^2 = x^2 + y^2$.

2. Compute the magnitude of both sides of the equation

   We take the magnitude of both sides of the given equation: $|x + iy| = \left|\sqrt{\frac{a+ib}{c+id}}\right|$

3. Simplify the magnitude equation

   Using the properties of magnitudes, we have: $|x + iy| = \sqrt{\left|\frac{a+ib}{c+id}\right|} = \sqrt{\frac{|a+ib|}{|c+id|}}$

4. Express magnitudes in terms of $a, b, c, d$

   Recall that $|a + ib| = \sqrt{a^2 + b^2}$ and $|c + id| = \sqrt{c^2 + d^2}$. So, $|x + iy| = \sqrt{\frac{\sqrt{a^2 + b^2}}{\sqrt{c^2 + d^2}}} = \left(\frac{a^2 + b^2}{c^2 + d^2}\right)^{1/4}$

5. Find $x^2 + y^2$

   We know that $|x + iy| = \sqrt{x^2 + y^2}$. Therefore, $x^2 + y^2 = |x + iy|^2$. So, $x^2 + y^2 = \left[\left(\frac{a^2 + b^2}{c^2 + d^2}\right)^{1/4}\right]^2 = \left(\frac{a^2 + b^2}{c^2 + d^2}\right)^{1/2}$

6. Compute $(x^2 + y^2)^2$

   Finally, we need to find $(x^2 + y^2)^2$. $(x^2 + y^2)^2 = \left[\left(\frac{a^2 + b^2}{c^2 + d^2}\right)^{1/2}\right]^2 = \frac{a^2 + b^2}{c^2 + d^2}$