Algebra: manipulation of expression How does this:

$$a\sqrt{2}=c$$

become this?:

$$\frac{a}{c}=\frac{a}{a\sqrt{2}}$$
If $c \neq 0$ you can divide both sides by $c$, you get:

$$\frac{a\sqrt{2}}{c}=\frac{c}{c}=1$$

Next divide both sides by $\sqrt{2}$

$$\frac{a}{c}=\frac{1}{\sqrt{2}}$$ 

You know that if $a \neq 0$ you have $\displaystyle 1=\frac{a}{a}$, so $\displaystyle \frac{1}{\sqrt{2}}=\frac{a}{a\sqrt{2}}$.
$$a\sqrt{2}=c$$

Divide both sides by $\sqrt{2}c$ ($c\ne0$) and clean up:

$$\frac{a\sqrt{2}}{\sqrt{2}c}=\frac{c}{\sqrt{2}c}$$

$$\frac{a}{c}=\frac{1}{\sqrt{2}}$$

By multiplying both numerator and denominator on the right hand side by $a$ ($a\ne0$) we get:

$$\frac{a}{c}=\frac{a}{a\sqrt{2}}$$