Let's solve the expression step-by-step.
The given expression is:
[tex]\[ \left(\frac{1}{27}\right)^2 \times 3^2 \times \frac{1}{3} \][/tex]
1. Rewrite the fractions as exponents of 3:
- [tex]\(\frac{1}{27}\)[/tex] can be written as [tex]\(3^{-3}\)[/tex], because [tex]\(27 = 3^3\)[/tex]. Therefore,
[tex]\[
\left( \frac{1}{27} \right)^2 = (3^{-3})^2.
\][/tex]
- For [tex]\(3^2\)[/tex], it remains the same.
- [tex]\(\frac{1}{3}\)[/tex] can be written as [tex]\(3^{-1}\)[/tex].
So, the expression now looks like:
[tex]\[ (3^{-3})^2 \times 3^2 \times 3^{-1} \][/tex]
2. Combine exponents where necessary:
- For [tex]\((3^{-3})^2\)[/tex], we can use the power of a power property [tex]\( (a^m)^n = a^{mn} \)[/tex]:
[tex]\[
(3^{-3})^2 = 3^{-3 \times 2} = 3^{-6}.
\][/tex]
Now, the expression is:
[tex]\[ 3^{-6} \times 3^2 \times 3^{-1} \][/tex]
3. Combine all terms with the same base:
- When multiplying exponents with the same base, add the exponents:
[tex]\[
3^{-6} \times 3^2 \times 3^{-1} = 3^{-6 + 2 - 1} = 3^{-5}.
\][/tex]
4. Evaluate the final expression:
- [tex]\(3^{-5}\)[/tex] means [tex]\(\frac{1}{3^5}\)[/tex].
Since [tex]\(3^5 = 243\)[/tex],
[tex]\[ 3^{-5} = \frac{1}{243}. \][/tex]
So, the final result of the expression [tex]\(\left(\frac{1}{27}\right)^2 \times 3^2 \times \frac{1}{3}\)[/tex] is:
[tex]\[ \frac{1}{243}, \][/tex]
which is approximately:
[tex]\[ 0.00411522633744856. \][/tex]
