What is this way of simplifying $\prod ^{n}_{i=1} x_i^k$ correct? I want the K out.

Question

What is the correct way of simplifying $\prod^{n}_{i=1} x_i^k$ ?

I tried getting out the k to have : $$ (\prod^{n}_{i=1} x_i)^{kn} $$

Is this correct?

Answer 1

No, it is not. It is a matter of reordering things:

$$ a_1^k \dots a_p^k = a_1 \dots a_1 \times \dots \times a_p \dots a_p = (a_1 \times \dots \times a_p ) \times \dots \times (a_1 \times \dots \times a_p ) = (a_1 \times \dots \times a_p )^k $$

So, $\prod_{i=1}^n x_i^k = (\prod_{i=1}^n x_i)^k$.