How to expand and solve using exponent laws and algebra?

Question

How do you solve this question, specifically between lines 2 and 3 in the image? The intuition, expanding and math behind it would be much appreciated!

Question needing solved

Answer 1

You have

$m\cdot i^{-1/2}=2\sqrt{m \cdot N\cdot 2\cdot i^{1/2}}$

Now we can replace the root sign by the exponent $1/2$.

$m\cdot i^{-1/2}=2(m \cdot N\cdot 2\cdot i^{1/2} )^{1/2}$

We know that $(x^b)^c=x^{b\cdot c}$

$m\cdot i^{-1/2}=2m^{1/2} \cdot N^{1/2}\cdot 2^{1/2}\cdot i^{1/4} $

Multiplying both sides by $i^{1/2}=i^{2/4}$

$m\cdot \underbrace{i^{-1/2}\cdot i^{1/2}}_{=1} =2\cdot m^{1/2} \cdot N^{1/2}\cdot 2^{1/2}\cdot i^{1/4+2/4}$

$m=2\cdot m^{1/2} \cdot N^{1/2}\cdot 2^{1/2}\cdot \color{blue}{ i^{3/4}}$

It is left to solve for $\color{blue}{ i^{3/4}}$. Can you proceed ?