1. ## Easy Product question

To express the following products in terms of $\pi$i=1 to n a_i where k is a constant :

$\Pi$i=1 to n ka_i...for this one would i just pull the constant k out so i would have k $\Pi$i=1 to n a_i to be in terms of $\Pi$i=1 to n a_i ?

and then I also have these two $\Pi$i=1 to n ia_i

$\Pi$i=1 to n (a_i)^k ..im not quite sure how to get it in terms of $\Pi$i=1 to n a_i.

2. Originally Posted by alice8675309
To express the following products in terms of $\pi$i=1 to n a_i where k is a constant :

$\Pi$i=1 to n ka_i...for this one would i just pull the constant k out so i would have k $\Pi$i=1 to n a_i to be in terms of $\Pi$i=1 to n a_i ?

and then I also have these two $\Pi$i=1 to n ia_i

$\Pi$i=1 to n (a_i)^k ..im not quite sure how to get it in terms of $\Pi$i=1 to n a_i.
Wow this is really hard to read. After staring for some time, I see you're trying to manipulate

$\displaystyle \prod_{i=1}^nka_i$

And you got

$\displaystyle k\prod_{i=1}^na_i$

which if you think about it will only be right if n=1; in general it's

$\displaystyle \prod_{i=1}^nka_i=k^n\prod_{i=1}^na_i$

Then you have another problem

$\displaystyle \prod_{i=1}^nia_i$

Question: have you learned factorials?

Lastly you have a question

$\displaystyle \prod_{i=1}^n(a_i)^k$

I don't know how to give a hint, just expand it and you will see that

$\displaystyle \prod_{i=1}^n(a_i)^k=\left(\prod_{i=1}^na_i\right) ^k$

You can visit the LaTeX Help subforum to learn about typesetting, and if you double click any rendered LaTeX on the site, it will display the code used to produce it.

3. Originally Posted by undefined
Wow this is really hard to read. After staring for some time, I see you're trying to manipulate

$\displaystyle \prod_{i=1}^nka_i$

And you got

$\displaystyle k\prod_{i=1}^na_i$

which if you think about it will only be right if n=1; in general it's

$\displaystyle \prod_{i=1}^nka_i=k^n\prod_{i=1}^na_i$

Then you have another problem

$\displaystyle \prod_{i=1}^nia_i$

Question: have you learned factorials?

Lastly you have a question

$\displaystyle \prod_{i=1}^n(a_i)^k$

I don't know how to give a hint, just expand it and you will see that

$\displaystyle \prod_{i=1}^n(a_i)^k=\left(\prod_{i=1}^na_i\right) ^k$

You can visit the LaTeX Help subforum to learn about typesetting, and if you double click any rendered LaTeX on the site, it will display the code used to produce it.
ok so Looking at and expanding the first one im good with that one and explaining it. the second one, $\displaystyle \prod_{i=1}^nia_i$ I started to expand it and it would be 1 $a_1$x2 $a_2$x...xn $a_n$ (the x is to indicate multiplication as opposed to a variable or anything else). Now, you asked if i've learned factorials yes, but as far as the number theory course briefly. Therefore, i can't wrap my head around it at the moment. and Finally, for the last one, it would expand out like this right? $(a_1)^k$x $(a_2)^k$x...x $(a_n)^k$..which is how you ended up with what you have right? And im sorry about my latex abilities, I have been referring to a chart but im not very good at it yet i need more practice with it.

4. Originally Posted by alice8675309
ok so Looking at and expanding the first one im good with that one and explaining it. the second one, $\displaystyle \prod_{i=1}^nia_i$ I started to expand it and it would be 1 $a_1$x2 $a_2$x...xn $a_n$ (the x is to indicate multiplication as opposed to a variable or anything else). Now, you asked if i've learned factorials yes, but as far as the number theory course briefly. Therefore, i can't wrap my head around it at the moment. and Finally, for the last one, it would expand out like this right? $(a_1)^k$x $(a_2)^k$x...x $(a_n)^k$..which is how you ended up with what you have right? And im sorry about my latex abilities, I have been referring to a chart but im not very good at it yet i need more practice with it.
Well multiplication is commutative, so

$\displaystyle 1a_1\cdot2a_2\cdots na_n = (1\cdot2\cdots n)(a_1a_2\cdots a_n) = n!\prod_{i=1}^na_i$

And yes $a_1^ka_2^k\cdots a_n^k=(a_1a_2\cdots a_n)^k$, this is just a basic property of exponents that you learn in algebra in grade school.