How do you simplify $\dfrac{{2{a^4}{b^3}}}{{{a^2}b}}?$

Hint: To simplify the given expression, first group similar terms (all similar variables in a group or parentheses and constant in other groups) with the help of commutative law of multiplication. Then after grouping the similar variables together and constants together, use law of indices for division to perform division between the similar variables and also perform algebraic operations (if exists in the given equation) with constants to simplify them too with the variable.

Complete step by step solution:
In order to simplify the given expression $\dfrac{{2{a^4}{b^3}}}{{{a^2}b}}$ we will first group similar terms in the given expression with the help of commutative property of multiplication as follows
Rewriting the given expression as $\dfrac{{2{a^4}{b^3}}}{{{a^2}b}}$ as
$ \Rightarrow 2{a^4}{b^3} \times \dfrac{1}{{{a^2}b}}$
Now, grouping similar terms, we will get
\[ \Rightarrow 2 \times \dfrac{{{a^4}}}{{{a^2}}} \times \dfrac{{{b^3}}}{b}\]
Now, we will use the law of indices for division in order to divide the similar variables present in the expression. Law of indices for division is given as when “x” to the power of “m” is being divided by “x” to the power of “n”, then their divide is given as “x” to the power of subtraction of “n” from “m”, mathematically it can be understood as follows
$\dfrac{{{x^m}}}{{{x^n}}} = {x^{m - n}}$
Using this for the above expression we will get
\[
   \Rightarrow 2 \times \dfrac{{{a^4}}}{{{a^2}}} \times \dfrac{{{b^3}}}{b} \\
   \Rightarrow 2 \times {a^{4 - 2}} \times {b^{3 - 1}} \\
   \Rightarrow 2 \times {a^2} \times {b^2} \\
   \Rightarrow 2{a^2}{b^2} \\
 \]
Therefore \[2{a^2}{b^2}\] is the simplified form of the expression $\dfrac{{2{a^4}{b^3}}}{{{a^2}b}}$

Note: We have written $\dfrac{{2{a^4}{b^3}}}{{{a^2}b}}\;{\text{as}}\;2{a^4}{b^3} \times \dfrac{1}{{{a^2}b}}$, in order to use the commutative property of multiplication to group the variables, since this property does not holds good for division and subtraction that’s why we have converted subtraction and multiplication with fraction to use this property.