Why does $\frac{1}{4}x^{-3/4} = \frac{1}{4x^{3/4}} = \frac{1}{4\sqrt[4]{x^3}}$? This is taken from Khan Academy, I don't understand how these equate:
$$\frac{1}{4}x^{-3/4}
= \frac{1}{4x^{3/4}}
= \frac{1}{4\sqrt[4]{x^3}}$$
How come the minus was remove from the original exponent?

These are properties of exponentiation. In particular,
$$a^{-b} = \frac{1}{a^b}$$
combined with
$$a^{\frac{m}{n}} = \sqrt[\leftroot{-2}\uproot{2}n]{a^m}.$$
In your case, 
$$\frac{1}{4}x^\frac{-3}{4} = \frac{1}{4x^\frac{3}{4}} = \frac{1}{4\sqrt[\leftroot{-2}\uproot{2}4]{x^3}}.$$
