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I've been trying learning Algebra, but have run into the expanding equations below. After trying for a while, I've come to the conclusion that I have no idea how to "fully" simplify something that expands with the following pattern:

1) $3^{a}-2^{a}$

2) $3^{a+c}-3^{c}*2^{a}+3^{c}*2^{a+b}-2^{a+b+c}$

3) $3^{a+c+e}-3^{c+e}*2^{a}+3^{c+e}*2^{a+b}-3^{e}*2^{a+b+c}+3^{e}*2^{a+b+c+d}-2^{a+b+c+d+e}$

I realize the above equations can be simplified a little from the current state, but I left it in this format because it is easier to see the pattern.

I ran into something like the above expansion before:

$3^{x-1}+3^{x-2}*2^{1}+3^{x-3}*2^{2}...+3^{2}*2^{x-3}+3^{1}*2^{x-2}+2^{x-1}$

Which ended up being equal to $3^{x}-2^{x}$ (which happens to be #1 above).

I am hoping to find a similar simplification for the expanding equations above. Can anyone point me in the direction of some literature on this kind of expansion?

Thank you!

JohnDNoone
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1 Answers1

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For 1) you can write $$3^a\left(1-\left(\frac{2}{3}\right)^a\right)$$ For 2) we have $$3^c\left(3^a-2^a+2^{a+b}\right)-2^{a+b+c}$$ For 3) I would write $$3^{c+e}(3^a-2^a+2^{a+b})-3^e\times 2^{a+b+c}(1-2^d)-2^{a+b+c+d+e}$$

Dr. Sonnhard Graubner
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