In expressions containing raised variables that are being divided, why do the variables not cancel each other out to a reciprocal of one? for example x^n/ x should be 1^n or is it still x^n?

In expressions containing raised variables that are being divided, why do the variables not cancel each other out to a reciprocal of one? for example x^n/ x should be 1^n or is it still x^n?

Relevant page
<a href="https://www.google.com/search?q=factoring+polynomials+with+exponents&oq=factoring+polynomials+with+exponents&aqs=chrome..69i57.14377j0j4&sourceid=chrome&ie=UTF-8">factoring polynomials with exponents - Google Search</a>
What I've done so far
(x^n/ x^n-2)^3 does this equal 1 or x^6?

It's often best to try some real numbers in algebraic expressions first, to see what is going on.

Let's try `10^4`.

This means `10 xx 10 xx 10 xx 10`

If we divide this by `10`, we would have

`(10 xx 10 xx 10 xx 10)/10`

We cancel one of the `10`s on top with the `10` on bottom, to give:

`10 xx 10 xx 10`

So what we've done is

`(10^4)/10 = 10^3`

If you do this for several other indices, you'll hopefully conclude that:

`(10^n)/10 = 10^(n-1)`

In general, for your first question, we'd have:

`x^n/ x = x^(n-1)`

For your other question, (x^n/ x^n-2)^3, I'm not sure if you mean `(x^n/ x^n-2)^3` or `(x^n/ (x^n-2))^3`?

You are encouraged to use the math entry system which makes it easier to express your questions and follow your working.

X

Hello Sam
It's often best to try some real numbers in algebraic expressions first, to see what is going on.
Let's try `10^4`.
This means `10 xx 10 xx 10 xx 10`
If we divide this by `10`, we would have
`(10 xx 10 xx 10 xx 10)/10`
We cancel one of the `10`s on top with the `10` on bottom, to give:
`10 xx 10 xx 10`
So what we've done is
`(10^4)/10 = 10^3`
If you do this for several other indices, you'll hopefully conclude that:
`(10^n)/10 = 10^(n-1)`
In general, for your first question, we'd have:
`x^n/ x = x^(n-1)`
For your other question, (x^n/ x^n-2)^3, I'm not sure if you mean `(x^n/ x^n-2)^3` or `(x^n/ (x^n-2))^3`?
You are encouraged to use the <a href="http://www.intmath.com/forum/entering-math-graphs-images-41/how-to-enter-math:91">math entry system</a> which makes it easier to express your questions and follow your working.

If his second question meant `(x^n/ x^n-2)^3`, the the result is `(1-2)^3=(-1)^3=-1`

If he meant `(x^n/ (x^n-2))^3`, then there is no "nice" simplification.

X

Sam didn't reply.
If his second question meant `(x^n/ x^n-2)^3`, the the result is `(1-2)^3=(-1)^3=-1`
If he meant `(x^n/ (x^n-2))^3`, then there is no "nice" simplification.