Simplify: sqrt(i / (3 - 4i))

To simplify the fraction, multiply the numerator and the denominator by the conjugate of the denominator, which is $3+4i$.

Since $i^2 = -1$:

So, the expression becomes $\sqrt{-\frac{4}{25} + \frac{3}{25}i}$.

Let $\sqrt{-\frac{4}{25} + \frac{3}{25}i} = x+yi$, where $x$ and $y$ are real numbers.
Squaring both sides:

Equating the real and imaginary parts, we get a system of equations:

1. $x^2 - y^2 = -\frac{4}{25}$
2. $2xy = \frac{3}{25}$

Also, the modulus of $x+yi$ squared is equal to the modulus of $-\frac{4}{25} + \frac{3}{25}i$:
$|x+yi|^2 = x^2+y^2$
$|-\frac{4}{25} + \frac{3}{25}i| = \sqrt{\left(-\frac{4}{25}\right)^2 + \left(\frac{3}{25}\right)^2} = \sqrt{\frac{16}{625} + \frac{9}{625}} = \sqrt{\frac{25}{625}} = \sqrt{\frac{1}{25}} = \frac{1}{5}$.
So, 3) $x^2 + y^2 = \frac{1}{5}$

Add equation (1) and (3):
$(x^2 - y^2) + (x^2 + y^2) = -\frac{4}{25} + \frac{1}{5}$
$2x^2 = -\frac{4}{25} + \frac{5}{25}$
$2x^2 = \frac{1}{25}$
$x^2 = \frac{1}{50}$
$x = \pm\sqrt{\frac{1}{50}} = \pm\frac{1}{5\sqrt{2}} = \pm\frac{\sqrt{2}}{10}$

Subtract equation (1) from (3):
$(x^2 + y^2) - (x^2 - y^2) = \frac{1}{5} - \left(-\frac{4}{25}\right)$
$2y^2 = \frac{5}{25} + \frac{4}{25}$
$2y^2 = \frac{9}{25}$
$y^2 = \frac{9}{50}$
$y = \pm\sqrt{\frac{9}{50}} = \pm\frac{3}{5\sqrt{2}} = \pm\frac{3\sqrt{2}}{10}$

From equation (2), $2xy = \frac{3}{25}$. Since $\frac{3}{25}$ is positive, $x$ and $y$ must have the same sign.
Therefore, the two possible square roots are:
$\frac{\sqrt{2}}{10} + i\frac{3\sqrt{2}}{10}$ and $-\frac{\sqrt{2}}{10} - i\frac{3\sqrt{2}}{10}$.

The principal square root of a complex number $z = a+bi$ is typically defined as the root with a positive real part, or if the real part is zero, the root with a positive imaginary part. In this case, the complex number inside the square root, $-\frac{4}{25} + \frac{3}{25}i$, has a negative real part and a positive imaginary part, placing it in the second quadrant. Its principal square root will have a positive real part and a positive imaginary part.