Can someone explain why the exponents behave like this

[treehau5]

Hello,

I can't seem to wrap my mind around this. I understand exponent properties, but for some reason when you throw that n in there it rocks my world.

I was solving an induction problem, and a piece of the algebra that I sort of guessed at was this:
(3ⁿ-3^n-1)

Which after factoring becomes
3^n-1(3-1)

I do not understand how the exponential division is working here with the n. Can someone please explain?

I tried testing it out a different way by just writing 3ⁿ/3^n-1 which equals 3, but this somehow confused me more.

 

[DonAntonio]

Hello,

I can't seem to wrap my mind around this. I understand exponent properties, but for some reason when you throw that n in there it rocks my world.

I was solving an induction problem, and a piece of the algebra that I sort of guessed at was this:
(3ⁿ-3^n-1)

Which after factoring becomes
3^n-1(3-1)

I do not understand how the exponential division is working here with the n. Can someone please explain?

I tried testing it out a different way by just writing 3ⁿ/3^n-1 which equals 3, but this somehow confused me more.

If you understand $\,xy-x=x(y-1)\,$, which is a simple application of the distributivity axiom (in some

field), then putting $\,x=3^{n-1}\,\,,\,y=3\,$ , we get:
$$3^n-3^{n-1}=3^{n-1}\cdot 3 -3^{n-1}=3^{n-1} (3-1)=2\cdot 3^{n-1}$$
Tadaaah!

DonAntonio