Simplify 6a(2a^2bc^3)

$6 a (2 a^{2} b c^{3})$

Multiply $a$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

$6 (a^{2} a) (2 b c^{3})$

Multiply $a^{2}$ by $a$ .

Raise $a$ to the power of $1$ .

$6 (a^{2} a^{1}) (2 b c^{3})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 a^{2 + 1} (2 b c^{3})$

Add $2$ and $1$ .

$6 a^{3} (2 b c^{3})$

Multiply $2$ by $6$ .

$12 a^{3} b c^{3}$