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?

• It's a basic rule of exponents. A negative in the exponent of a term in the numerator appears as a positive exponent of the term put in the denominator: $x^{-n}=\frac{1}{x^n}$ Apr 11, 2016 at 3:33

These are properties of exponentiation. In particular,

$$a^{-b} = \frac{1}{a^b}$$

combined with

$$a^{\frac{m}{n}} = \sqrt[\leftroot{-2}\uproot{2}n]{a^m}.$$

$$\frac{1}{4}x^\frac{-3}{4} = \frac{1}{4x^\frac{3}{4}} = \frac{1}{4\sqrt[\leftroot{-2}\uproot{2}4]{x^3}}.$$

• These properties have an intuitive ring to them: if raising something to a positive exponent is repeated multiplication, it is not unreasonable to consider raising something to a negative exponent as repeated division. Going beyond this intuitive pattern, one finds that all rules of exponents which work for positive exponents work for negative ones as well, if you define a negative exponent in this way. As an example, a^(x-y) = a^x/a^y. Apr 11, 2016 at 5:19