Proper use of the neg. exponent?

[Silvanoshei]

Lets use \(\displaystyle \frac{3}{x}\), if you were to give it a neg. exponent it'd be \(\displaystyle x^{-3}\)?

Or if we had \(\displaystyle \frac{x}{3}\)? That can't be done? Or I'm I reversing this?

 

[pka]

Lets use \(\displaystyle \frac{3}{x}\), if you were to give it a neg. exponent it'd be \(\displaystyle x^{-3}\)?

Or if we had \(\displaystyle \frac{x}{3}\)? That can't be done? Or I'm I reversing this?

\(\displaystyle \dfrac{3}{x} = 3{x^{ - 1}}\;\& \;\dfrac{x}{3} = {3^{ - 1}}x\)

 

[Silvanoshei]

\(\displaystyle \dfrac{3}{x} = 3{x^{ - 1}}\;\& \;\dfrac{x}{3} = {3^{ - 1}}x\)

Whats this operation called? Reference?

So if you had \(\displaystyle \frac{3}{x^{2}}\).... it'd be \(\displaystyle 3x^{-2}\)?

 

[JeffM]

Whats this operation called? Reference?

So if you had \(\displaystyle \frac{3}{x^{2}}\).... it'd be \(\displaystyle 3x^{-2}\)? YES

It's not really an operation: it's a definition

\(\displaystyle a \ne 0 \implies a^{-b} \equiv \dfrac{1}{a^b} \equiv \left(\dfrac{1}{a}\right)^b.\)