To calculate: The simplified value of the radical expression displaystylesqrt{{{18}{a}}}divsqrt{{{2}{a}^{4}}}.

Given expression is ${\displaystyle \sqrt{18a}÷\sqrt{2a^4}.}$
Formula used:
The product rule for radicals,
${\displaystyle \sqrt[n]{a}b=\sqrt[n]{a}\cdot \sqrt[n]{b}}$
Here, n is the positive integer and a and b are real number.
The relation between radical and rational exponent notation is expressed as,
${\displaystyle \sqrt[n]{a}=a^{1\text{/}n}.}$
Here, a is the radicand, and n is the index of the radical.
Calculation:
Consider the given expression,
${\displaystyle \sqrt{18a}÷\sqrt{2a^4}.}$
Use the product rule for radicals to simplify the expression,
${\displaystyle \sqrt{18a}÷\sqrt{2a^4}=\frac{\sqrt{18a}}{\sqrt{2a^4}}}$
${\displaystyle =\frac{\sqrt{9\cdot 2\cdot a}}{\sqrt{2}\cdot a^2\cdot a^2}}$
${\displaystyle =\frac{\sqrt{3^2}\cdot \sqrt{2}\cdot \sqrt{a}}{\sqrt{2}\cdot \sqrt{a^2\cdot \sqrt{a^2}}}}$
Use the relation between radical and rational exponent notation to simplify the expression,
${\displaystyle \sqrt{18a}÷\sqrt{2a^4}=\frac{{\left(3^2\right)}^{1\text{/}2}\sqrt{a}}{{\left(a^2\right)}^{1\text{/}2}{\left(a^2\right)}^{1\text{/}2}}}$
${\displaystyle =\frac{{\left(3\right)}^{2\left(1\text{/}2\right)}\sqrt{a}}{{\left(a\right)}^{2\left(1\text{/}2\right)}{\left(a\right)}^{2\left(1\text{/}2\right)}}}$
${\displaystyle =\frac{\left(3\right)\sqrt{a}}{\left(a\right)\left(a\right)}}$
${\displaystyle =\frac{3\sqrt{a}}{n^2}}$
Hence, the simplified value of the provided expression is ${\displaystyle \frac{3\sqrt{a}}{a^2}.}$