The simplified form of the expression displaystyle{sqrt[{{4}}]{{{c}{d}^{2}}}}times{sqrt[{{3}}]{{{c}^{2}{d}}}}.

Formula used:
Product property of radicals:
If a and b are real numbers and $n>1$ is an integer, the product property is true provided that the radicals are real numbers.
${\displaystyle \sqrt[n]{a}\times \sqrt[n]{b}=\sqrt[n]{ab}}$
If m and n are integers and $n>1$ is an integer, then
${\displaystyle \sqrt[n]{a^m}=a^{mn}}$
Calculation:
Consider the expression,
${\displaystyle \sqrt[4]{c{d}^2}\times \sqrt[3]{c^{2}d}}$
Rewrite the provided expression as rational exponents.
${\displaystyle \sqrt[4]{c{d}^2}\times \sqrt[3]{c^{2}d}={\left(c{d}^2\right)}^{1\text{/}4}\times {\left(c^{2}d\right)}^{1\text{/}3}}$
Use the product property.
${\displaystyle {\left(c{d}^2\right)}^{1\text{/}4}\times {\left(c^{2}d\right)}^{1\text{/}3}={\left(c\right)}^{1\text{/}4}\times {\left(d^2\right)}^{1\text{/}4}\times {\left(c^2\right)}^{1\text{/}3}\times {\left(d\right)}^{1\text{/}3}}$
Use the formula ${\displaystyle \sqrt[n]{a^m}=a^{m\text{/}n}}$ and simplify the expression.
${\displaystyle {\left(c\right)}^{1\text{/}4}\times {\left(d^2\right)}^{1\text{/}4}\times {\left(c^2\right)}^{1\text{/}3}\times {\left(d\right)}^{1\text{/}3}=c^{1\text{/}4}\times d^{2\text{/}4}\times c^{2\text{/}3}\times d^{1\text{/}3}}$
Add the powers of the same bases.
${\displaystyle c^{1\text{/}4}\times d^{2\text{/}4}\times c^{2\text{/}3}\times d^{1\text{/}3}=c\left(\frac{1}{4}\right)+\left(\frac{2}{4}\right)\times d\left(\frac{2}{4}\right)+\left(\frac{1}{3}\right)}$
${\displaystyle =c^{\frac{11}{12}}\times d^{\frac{10}{12}}}$
The obtained expression with rational exponents can be rewritten into radicals as,
${\displaystyle =c^{\frac{11}{12}}\times d^{\frac{10}{12}}={\left(c^{11}\right)}^{\frac{1}{12}}\times {\left(d^{10}\right)}^{\frac{1}{12}}}$
${\displaystyle =\sqrt[12]{c^{11}{d}^{10}}}$
Answer: ${\displaystyle \sqrt[4]{c{d}^2}\times \sqrt[3]{c^{2}d}is\sqrt[12]{c^{11}{d}^{10}}}$