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It is given that ${{3}^{x}}=6561$, now we have to calculate the prime factorization of 6561.

"Prime Factorization" is finding which prime numbers multiply together to make the original number.

Now we have the number 6561.

So, $6561=3\times 3\times 3\times 3\times 3\times 3\times 3\times 3$

It can also be written as, $6561={{3}^{8}}$.

Putting the value of 6561 in the form of prime factorization, we get

\[\therefore {{3}^{x}}={{3}^{8}}\]

Comparing the powers of 3 both sides, we get

\[\Rightarrow x=8\]

Hence, the value of $x$ is equal to 8.

Now, we have to find the value of ${{3}^{x-3}}$, so putting the value of $x$, we get

$\therefore {{3}^{x-3}}={{3}^{8-3}}={{3}^{5}}=243$

We have, ${{3}^{x}}=6561$

It can also be written as, \[{{3}^{x}}={{3}^{8}}\]

Now, dividing both sides by ${{3}^{3}}$, we get

$\Rightarrow \dfrac{{{3}^{x}}}{{{3}^{3}}}=\dfrac{{{3}^{8}}}{{{3}^{3}}}$

$\Rightarrow {{3}^{x-3}}={{3}^{8-3}}={{3}^{5}}$

$\Rightarrow {{3}^{x-3}}=243$

Hence, the answer is 243.