**finally,i'm able to write on forum **
Hi, First of all, I would like to thank you for taking your timeâ€¦â€¦

Prove that: $(1-Ï‰)(1-Ï‰^2)(1-Ï‰^4)(1-Ï‰^8$)â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦to 2n factors $=3^n$

In my problems book ,I found this problem. I tried to solve it by myself â€œas usual I canâ€™t solve it at first time â€œ

. so I saw my guide answer book . the answer wasnâ€™t clear for me at first time â€œthat is unusual â€ I asked my teacher for that question but he didnâ€™t add anything for me to understand it. so, I search on the internet for explanation but most of them are repeat what my book say !! I donâ€™t lose my hope. I found someone asked the same question in this site

https://math.stackexchange.com/questions/2299230/what-is-meaning-of-2n-factors
From the answers on this question I knew The 2n factors mean theâ€ numbers of factors â€œ but I wrote it in my book as a note .

I continue my study normally but my brain canâ€™t stop thinking about that problem

So, I decide to break up my problem â€¦â€¦â€¦â€¦..

I note $(1-Ï‰)(1-Ï‰^2)(1-Ï‰^4)$ they are factors multiply to each other

I began to search â€œhow to multiply factorsâ€â€¦â€¦.after 4 hours

I found that method to multiply factors

https://www.youtube.com/watch?v=g4zTQveQuuY
**$N^{no.of factors/2}$**
Number of factors =2n â€œme: oh yeah!!!â€

What is n ??

we should study factors first

"Factors" are the numbers we can multiply together to get another number:

$2\times{3}$ $ =6$

2 and 3 are factors of 6

A number can have many factors.

For example :

$12 =$ $1\times{12}$

$= $ $2\times6$

$=$ $3\times4$

so all factors of â€œ12â€ ={1,2,3,4,6,12} note itâ€™s set.set should have all unique items. We canâ€™t repeat numbers

now I will tell you â€œnâ€=12 â€œthe number that has factorsâ€

if we multiply these factors together we will find =1728

letâ€™s use the formula $N^{no.of factors/2}$

we have 6 factors of 12

$12^{6/2}$ $=1728$

Then, the formula is valid

Before we solve our problem remember these relation to help us simplify our problem

$Ï‰^3$=$1$

$1+Ï‰+$$Ï‰^2$$=0$

$-Ï‰=1+$$Ï‰^2$

$-Ï‰^2$=$1+Ï‰$

But , wait how to get â€œnâ€ from our problem . we know we get it by multiply two factors

But we canâ€™t get 12 by multiply 1 and 2 â€œ1&2 are factors of 12â€ we should find the right two factors to multiply to each otherâ€¦..okay!

**Letâ€™s solve our problem from this point **
Prove that: $(1-Ï‰)(1-Ï‰^2)(1-Ï‰^4)(1-Ï‰^8$)â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦to 2n factors $=3^n$

$(1-Ï‰)$ will equal $(1+1+$$Ï‰^2)$$=(2+$$Ï‰^2)$

$(1-Ï‰^2)=(1+1+Ï‰)=(2+ Ï‰)$

The next bracket we will find $Ï‰^4=Ï‰^3\times Ï‰=1\times Ï‰=Ï‰$

$(1-Ï‰^4)=(1-Ï‰)=(1+1+Ï‰^2)=(2+Ï‰^2)$

So letâ€™s put them front of us

$(2+Ï‰^2) (2+ Ï‰) (2+Ï‰^2)$â€¦â€¦etc. we will note that two bracket repeatâ€¦..â€two factors repeatâ€

So we have only two factors $(2+Ï‰^2 )(2+Ï‰)$ which they are equal to â€œnâ€â€¦â€¦â€¦â€¦..and number of factors is â€œ2nâ€

Letâ€™s use multiply factors formula $N^{no.of factors/2}$

$((2+Ï‰^2)(2+Ï‰))^{2n/2}$ â€¦â€¦â€¦â€¦2 will go away with 2

$((2+Ï‰^2)(2+Ï‰))^n=(4+2Ï‰+2Ï‰^2+Ï‰^3)^n=(4+2Ï‰+2 Ï‰^2+1)^n=(2+2+2Ï‰+2Ï‰^2+1)^n=(2+2(1+Ï‰+Ï‰^2)+1)^n$

$=3^n$

Thanks $\color{red}{â™¥â™¥}$

Yahia kamal 2019