Question
21. If \( 3^{x}=81 \), then find the value of \( 3^{x-6} \)
Ask by May Washington. in India
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( 3^{x-6} = \frac{1}{9} \)
Solution
To solve the equation \( 3^{x} = 81 \), we first express 81 as a power of 3.
We know that:
\[
81 = 3^4
\]
So we can rewrite the equation as:
\[
3^{x} = 3^4
\]
Since the bases are the same, we can set the exponents equal to each other:
\[
x = 4
\]
Now, we need to find the value of \( 3^{x-6} \). Substituting \( x = 4 \) into the expression gives us:
\[
3^{x-6} = 3^{4-6} = 3^{-2}
\]
Now we can calculate \( 3^{-2} \):
\[
3^{-2} = \frac{1}{3^2} = \frac{1}{9}
\]
Thus, the value of \( 3^{x-6} \) is:
\[
\frac{1}{9}
\]
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Simplify this solution Beyond the Answer
To solve \( 3^{x}=81 \), we can rewrite 81 as a power of 3. Notice that \( 81 = 3^{4} \). Therefore, we can set the exponents equal to each other: \[ x = 4. \] Now, we want to find the value of \( 3^{x-6} \). Substituting \( x \) into the expression gives us: \[ 3^{x-6} = 3^{4-6} = 3^{-2}. \] Since \( 3^{-2} = \frac{1}{3^{2}} = \frac{1}{9} \), we find that: \[ 3^{x-6} = \frac{1}{9}. \] Thus, the answer is: \[ \frac{1}{9}. \]
