Simplify an expression with negative exponents

[dkonnen]

Okay the expression is

2r(n-1)r^(n-1) - (r^2)n(n-2)r^(n-2)

I'm confused after this point. What we did in class was move the negative exponent to the denominator. So I got:

2r(n-1)r^n over r for the first expression minus (r^2)n(n-2)r^n over r^2 for the second expression

Then I eliminated like terms, so I scratched the r from the denominator in the first term and the r in the 2r part of the numerator so the first term became:

2(n-1)r^n

Then I eliminated like terms in the second expression, so I scratched out the r^2 in the numerator and denominator to get:

n(n-2)r^n

So the term I have now is 2(n-1)r^n - n(n-2)r^n

Am I on the right track and is this as simple as I can make it?

 

[Denis]

So the term I have now is 2(n-1)r^n - n(n-2)r^n
Am I on the right track and is this as simple as I can make it?

Good work!

You can simplify a bit further; take the r^n out:
r^n[2(n-1) - n(n-2)]
Next, simplify what's inside the square brackets; OK?

 

[soroban]

Hello, dkonnen!

If I'm reading it correctly, I don't see any negative exponents.

\(\displaystyle 2r(n-1)r^{n-1}\:-\:r^2n(n-2)r^{n-2}\)

The first term is: \(\displaystyle \:2(n-1)\,\cdot\,r\,\cdot\,r^{n-1} \;=\;2(n-1)r^n\)

The second term is: \(\displaystyle \:n(n-2)\,\cdot\,r^2\,\cdot\,r^{n-2} \;=\;n(n-2)r^n\)


The problem becomes: \(\displaystyle \:2(n-1)\,\cdot\,r^n\:-\:n(n-2)\,\cdot\,r^n\)

Factor: \(\displaystyle \:r^n[2(n-1)\,-\,n(n-2)]\;=\;r^n[2n\,-\,2\,-\,n^2\,+\,2n] \;=\;-r^n(n^2\,-\,4n\,+\,2)\)