According to the given question,
Product of \[{{3}^{rd}}~and\text{ }{{8}^{th}}\] terms of a G.P. is \[243\]
The general term of a G.P.
First term \[a\]
And
Common ratio \[r\]is given by,
\[{{t}_{n}}~=\text{ }a{{r}^{n\text{ }-\text{ }1}}\]
So,
\[{{t}_{3}}~x\text{ }{{t}_{8~}}=\text{ }a{{r}^{3\text{ }-\text{ }1}}~x\text{ }a{{r}^{8\text{ }-\text{ }1}}\]
\[=\text{ }a{{r}^{2}}~x\text{ }a{{r}^{7~}}=\text{ }{{a}^{2}}{{r}^{9}}~=\text{ }243\]
Also,
\[{{t}_{4}}~=\text{ }a{{r}^{4\text{ }-\text{ }1}}~=\text{ }a{{r}^{3}}~=\text{ }3\]
Now,
\[{{a}^{2}}{{r}^{9}}~=\text{ }(a{{r}^{3}})\text{ }(a{{r}^{6}})~=\text{ }243\]
Substituting the value of \[a{{r}^{3}}\], we get
\[\left( 3 \right)\text{ }a{{r}^{6}}~=\text{ }243\]
\[a{{r}^{6}}~=\text{ }81\]
\[a{{r}^{7\text{ }-\text{ }1}}~=\text{ }81\text{ }=\text{ }{{t}_{7}}\]
So, the \[{{7}^{th}}~\]term of G.P. is \[81.\]