How do you simplify ${{\left( 2{{x}^{2}}{{y}^{3}} \right)}^{2}}$ ?

Hint:First try to separate the variables by product distribution rule. The power will also be distributed for each variable and will be raised by that power. Then for the power over power, multiply the powers to get the simplified value.

Complete step-by-step answer:
Starting with our expression ${{\left( 2{{x}^{2}}{{y}^{3}} \right)}^{2}}$
As we know ${{\left( ab \right)}^{n}}={{a}^{n}}{{b}^{n}}$,
So using this formula to distribute the exponent, we get
$={{2}^{2}}{{\left( {{x}^{2}} \right)}^{2}}{{\left( {{y}^{3}} \right)}^{2}}$ (by applying product distribution rule for $2{{x}^{2}}{{y}^{3}}$, the power of 2 ,${{x}^{2}}$and ${{y}^{3}}$are raised by the power ‘2’ .)
Now we have to take care of ‘power over power’ part as in ${{\left( {{x}^{2}} \right)}^{2}}$ and ${{\left( {{y}^{3}} \right)}^{2}}$
Since , we know ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$,
So using this formula to multiply the exponents, we get
$=4{{x}^{2\cdot 2}}{{y}^{3\cdot 2}}$ (by applying power rule and multiplying the powers of the variables ‘x’ and ‘y’ .)
$=4{{x}^{4}}{{y}^{6}}$
Hence we can write ${{\left( 2{{x}^{2}}{{y}^{3}} \right)}^{2}}=4{{x}^{4}}{{y}^{6}}$
This is the simplified form of the given expression.

Note: Variable should be separated first, so that the power would raise for every individual variable. For ‘power over power’ of variables, the powers should be multiplied as per the rule.