Merging similar topics.

GSDster wrote:

I'm using the

MGMAT Number Properties book and I'm hoping someone can please explain a concept for me.

Going to and from Distributed Form and Factored Form...

a^b-a^(b-1) turns into a^b(1-a^(-1)) = a^(b-1)(a-1)

I understand the first step, but how do you get from a^b(1-a^(-1)) to a^(b-1)(a-1)?

Sorry for not using the math tag. The equation in the exponent seems to mess up the formatting. Thanks in advance!

The only difference is the names of variables:

You can directly factor out \(a^{b-1}\) from \(a^b-a^{b-1}\) and get \(a^{b-1}*(a-1)\) as \(a^{b-1}*(a-1)=a^{b-1}*a-a^{b-1}=a^b-a^{b-1}\).

Or the long way: \(a^b-a^{b-1}=a^b-a^b*a^{-1}\) -> factor out \(a^b\): \(a^b(1-a^{-1})=a^b(1-\frac{1}{a})=a^b(\frac{a-1}{a})=\frac{a^b}{a}*(a-1)=a^{b-1}*(a-1)\).

Thanks for the response! Man, that long way really is long. Are there any times when I will actually have to do the long way, or can I just sort of skip all that and memorize the format of the factored form? I guess what I'm really asking is, are there any conditions that I should be aware of for when I can or can't use the shortcut?