Simplify: (3x2y)(2xy3) (b) Express your answer in index form

Using the rule $x^a \cdot x^b = x^{a+b}$:

$$x^2 \cdot x^1 = x^{2+1} = x^3$$

Using the rule $y^a \cdot y^b = y^{a+b}$:

$$y^1 \cdot y^3 = y^{1+3} = y^4$$

$$(3x^2y)(2xy^3) = 6x^3y^4$$

The simplified expression is $\boxed{6x^3y^4}$.

12. (b) Express your answer in index form

The simplified expression from part (a) is already in index form, where variables are written with their exponents.
The answer in index form is $\boxed{6x^3y^4}$.