Question

Consider the following equation, ${{3}^{x}}=6561$, then ${{3}^{x-3}}$ is?

Answer 1

Hint: In order to solve this question we have to first calculate the prime factorization of 6561 and then represent it in powers of 3. After representing, compare the powers of 3 both sides and get a linear equation in $x$. Get the value of $x$ after solving the equation and put it in place of $x$ and find ${{3}^{x-3}}$.

Complete step-by-step solution:
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
$\therefore {{3}^{x-3}}={{3}^{8-3}}={{3}^{5}}=243$
So, the answer of the question is 243 i.e. ${{3}^{x-3}}=243$.

Note: In this type of question, the crux lies in representing the number in RHS in powers of prime numbers, which can be done easily by representing the prime factorization of the number in RHS. After representing in this way, compare the powers and get the value of the unknown variable. We can also solve this question with a few different methods.
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.