# Thread: Sum or Difference between any powers.

1. ## Sum or Difference between any powers.

I understand the difference between two squares but I don't understand this.

$\displaystyle a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$

2. Originally Posted by alyosha2
I understand the difference between two squares but I don't understand this.

$\displaystyle a^5 - a^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$
I would call that 0.

3. Originally Posted by dwsmith
I would call that 0.

no I mean I don't understand the process that led to the difference of those two terms being displayed in the way that it is on the right.

4. I think you mean $\displaystyle \displaystyle a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$

Start with $\displaystyle \displaystyle (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$ expand it out, what do you get?

Can you guess the following?

$\displaystyle \displaystyle a^3 - b^3 = \dots$

$\displaystyle \displaystyle a^4 - b^4 = \dots$

5. Originally Posted by alyosha2
no I mean I don't understand the process that led to the difference of those two terms being displayed in the way that it is on the right.
Technically the RHS is 0 since $\displaystyle a^5-a^5=0$

6. Originally Posted by dwsmith
Technically the RHS is 0 since $\displaystyle a^5-a^5=0$
sorry, my mistake. I meant the second a to be a b.

7. When you say expand do you mean factor out?

8. Originally Posted by pickslides
I think you mean $\displaystyle \displaystyle a^5 - b^5 = (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$

Start with $\displaystyle \displaystyle (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$ expand it out, what do you get?

Can you guess the following?

$\displaystyle \displaystyle a^3 - b^3 = \dots$

$\displaystyle \displaystyle a^4 - b^4 = \dots$
when you say expand out do you mean factor out?

9. Spoiler:
$\displaystyle \displaystyle a^n - b^n = (a - b)(a^{n-1}b^0 + a^{n-2}b^1 + \dots +a^0 b^{n-1})$

10. Expand $\displaystyle \displaystyle (a - b)(a^4 + a^3b + a^2b^2 + ab^3 + b^4)$ and simplify. Does it equal $\displaystyle \displaystyle a^5 - b^5$?

Can you guess the factorisations of $\displaystyle \displaystyle a^3 - b^3$ and $\displaystyle \displaystyle a^4 - b^4$? Can you see a pattern?

11. Originally Posted by alyosha2
when you say expand out do you mean factor out?
I'm asking you to multiply the brackets out. There is a pattern here, can you find it?

12. Here is what I understand.

$\displaystyle a^5 - b^5 = (a + b)(a^4 - b^4) = (a + b)(a + b)(a^3 - a^3) = (a + b)(a + b)(a + b)(a^2 - b^2) = (a + b)(a + b)(a + b)(a + b)(a - b)$

but I don't understand how we get to the right of what I first posted.

13. $\displaystyle a^5-b^5 \neq (a+b)(a^4-b^4)$

Prove it! and I are suggesting you start with the RHS to show the LHS, maybe this might clear some things up.

14. Originally Posted by pickslides
I'm asking you to multiply the brackets out. There is a pattern here, can you find it?
I can see the pattern but I don't understand how we get to this way of expressing the difference between the first two powers.

15. I get to

$\displaystyle a^5b + a^4b^2 + a^3b^3 + a^2b^4 + ab^5$

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