How Do You Convert \( \frac{1}{x^n} \) to \( x^{-n} \) in Algebra?

[BkBkBk]

ok so I am working through a problem,got part way through and realized i couldn't remeber what to do with power rules,if someone could quickly remind me of the rules!

the question is differentiate

ƒ(x) = 1/x^3

Using

if ƒ(x) = x^n then ƒ'(x) = nx^n-1

and the book answer given is
ƒ(x) = 1/x^3 = x^-3

so the derivative is ƒ'(x) = -3x^-4

now i can understand all but one part of this

the part which is how to get from

1/x³ to x^-³

or why

1/xⁿ becomes x-ⁿ

i remember is one of the basic power laws,just cannot remember how or why its so.
could someone remind me please!

 

[The Chaz]

Part a. x^n * 1/x^n = 1.
To see this, notice that the left hand member is the same as
x^n/x^n, and any nonzero number divided by itself is equal to one.
Part b. x^n * x^-n = 1.
To see this, use the rule of exponents that says "when multiplying like bases, add the exponents". So you will have x^(n + -n), which is equal to x^(n - n) = x^0 = 1.
Now look at what changed from part a to part b. I replaced the "1/x^n" in part a with "x^-n" in part b, and the result was unchanged, so the two expressions are equal.

A more formal approach (using inverse operations, etc) will probably be posted if that's what you're looking for.

 

[Rasalhague]

The notations are equivalent:

$$b^{-p}\equiv\frac{1}{b^p}$$

The left hand side and the right hand side of this equation are two different ways of writing the same thing.

 

[Rasalhague]

For the field of real numbers with addition and multiplication as usually defined:

$$b^{p+q}=b^pb^q$$

$$b^{pq}=(b^p)^q$$

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

$$b^{p+(-q)}=b^{p-q}=\frac{b^p}{b^q}=b^p(b^q)^{-1}=b^pb^{-q}$$

$$b^{p/q}=\sqrt[q]{b^p}$$

where -x means the inverse of x with respect to the operation of addition (the additive inverse of x), so that x-x means x+(-x) = 0, and 1/x = x^-1 means the inverse of x with respect to the operation of multiplication (the multiplicative inverse of x), so that x/x = xx^-1 = 1. These rules are general except that 0^-1 (and hence any negative power of 0) is not defined.

If p and q are whole numbers, b^p can be interpreted as multiply p b's together, and b^p-q multiply p b's together with q instances of the multiplicative inverse of b, with the convention that b⁰ = 1. The rest of the real numbered powers "fill in the gaps" between rational powers (fractional powers). I don't yet know how that filling in of the gaps is formally defined, but I'm sure there are lots of people here who do!