How can you rewrite powers like this: a^x - b^x?

MaryB · #1 $Old$ November 13th, 2009, 04:15 AM

Hi there,
I am proving something, and I want to know how to rewrite a formula like this one: a^x - b^x, where you know that b<a.
Is there any way to write this differently? Maybe with sumformula, or an easy powerrule...anything?

HallsofIvy · #2 $Old$ November 13th, 2009, 04:57 AM

Quote:

Originally Posted by MaryB $View Post$

Hi there,
I am proving something, and I want to know how to rewrite a formula like this one: a^x - b^x, where you know that b<a.
Is there any way to write this differently? Maybe with sumformula, or an easy powerrule...anything?

There is no "rule of exponents" that will simplify that.

IF x is a positive integer, then $a^x- b^x= (a- b)(a^{x-1}+ a^{x-2}b+ a^{x-3}b^2+ \cdot\cdot\cdot+ a^2b^{x-3}+ ab^{x-2}+ b^{x-1})$ but I doubt that is what you want.

MaryB · #3 $Old$ November 13th, 2009, 05:42 AM

Actually, that is a very useful rewriting you gave me there.
Does it have a name? Or is it deduced from another formula?
I think I can use it!

Moo · #4 $Old$ November 13th, 2009, 09:45 AM

Hello,

See here : a^k - b^k = for a way to prove it

MaryB · #5 $Old$ November 13th, 2009, 10:26 AM

Quote:

Originally Posted by Moo $View Post$

Hello,
See here : a^k - b^k = for a way to prove it

Brilliant thanks! Btw, I love your avatar.
Okay well, my assignment is getting near the solution, but not yet. So I now have $(a- b)(a^{x-1}+ a^{x-2}b+ a^{x-3}b^2+ \cdot\cdot\cdot+ a^2b^{x-3}+ ab^{x-2}+ b^{x-1})$ and found out that if a = a+1, then b = a. so...now I have $(a+1 -a)((a+1)^{x-1}+ (a+1)^{x-2}a+ (a+1)^{x-3}a^2+ \cdot\cdot\cdot+ (a+1)^2a^{x-3}+ (a+1)a^{x-2}+ a^{x-1})$
And yeah, okay. How can I rewrite this one? I don't think it's neccesary to give the whole rewriting, but a hint? (Or the whole rewriting is okay, I can stop myself from looking first and try it out myself).

MaryB · #6 $Old$ November 14th, 2009, 03:37 AM

Quote:

Originally Posted by MaryB $View Post$

$(a+1 -a)((a+1)^{x-1}+ (a+1)^{x-2}a+ (a+1)^{x-3}a^2+ \cdot\cdot\cdot+ (a+1)^2a^{x-3}+ (a+1)a^{x-2}+ a^{x-1})$
And yeah, okay. How can I rewrite this one? I don't think it's neccesary to give the whole rewriting, but a hint? (Or the whole rewriting is okay, I can stop myself from looking first and try it out myself).

Does anyone know if I can rewrite in a way it shows a sumformula? Because I think that is quite possible, but I don't really know..how!
Please help