Calculation:

To find the product of given expression, we will use distributive property

\(\displaystyle{a}{\left({b}+{c}\right)}={a}{b}+{a}{c}\) as shown below:

\(\displaystyle\sqrt{{3}}{\left(\sqrt{{27}}-\sqrt{{3}}\right)}\) (Given expression)

\(\displaystyle\Rightarrow\sqrt{{3}}\times\sqrt{{27}}-\sqrt{{3}}\times\sqrt{{3}}\) (Using distributive property)

\(\displaystyle\Rightarrow\sqrt{{{3}\times{27}}}-\sqrt{{{3}\times{3}}}\) (Applying rule root(n)a xx root(n)b = root(n)(ab))

\(\displaystyle\Rightarrow\sqrt{{81}}-\sqrt{{9}}\)

\(\displaystyle\Rightarrow\sqrt{{{9}^{2}}}-\sqrt{{{3}^{2}}}\)

\(\displaystyle\Rightarrow{9}-{3}\ \text{(Applying rule}\ \displaystyle{\sqrt[{{n}}]{{{a}^{n}}}}={a}\))

\(\displaystyle\Rightarrow{6}\)

Therefore, the product of the given expressions would be 6.

To find the product of given expression, we will use distributive property

\(\displaystyle{a}{\left({b}+{c}\right)}={a}{b}+{a}{c}\) as shown below:

\(\displaystyle\sqrt{{3}}{\left(\sqrt{{27}}-\sqrt{{3}}\right)}\) (Given expression)

\(\displaystyle\Rightarrow\sqrt{{3}}\times\sqrt{{27}}-\sqrt{{3}}\times\sqrt{{3}}\) (Using distributive property)

\(\displaystyle\Rightarrow\sqrt{{{3}\times{27}}}-\sqrt{{{3}\times{3}}}\) (Applying rule root(n)a xx root(n)b = root(n)(ab))

\(\displaystyle\Rightarrow\sqrt{{81}}-\sqrt{{9}}\)

\(\displaystyle\Rightarrow\sqrt{{{9}^{2}}}-\sqrt{{{3}^{2}}}\)

\(\displaystyle\Rightarrow{9}-{3}\ \text{(Applying rule}\ \displaystyle{\sqrt[{{n}}]{{{a}^{n}}}}={a}\))

\(\displaystyle\Rightarrow{6}\)

Therefore, the product of the given expressions would be 6.