To determine which expression is equivalent to [tex]\(81^{\frac{1}{3}}\)[/tex], we can analyze each of the given options and simplify them step by step.
First, we know that the answer to [tex]\(81^{\frac{1}{3}}\)[/tex] is approximately 4.3267487109222245, so we'll look for an expression that simplifies to this number.
1. Option [tex]\(3 \sqrt[3]{3}\)[/tex]:
[tex]\[
3 \cdot \sqrt[3]{3}
\][/tex]
We know that [tex]\(\sqrt[3]{3}\)[/tex] is approximately 1.4422, so:
[tex]\[
3 \cdot 1.4422 \approx 4.3266
\][/tex]
This option simplifies to a number very close to 4.3267487109222245.
2. Option [tex]\(3 \sqrt{3^3}\)[/tex]:
[tex]\[
3 \cdot \sqrt{3^3} = 3 \cdot \sqrt{27}
\][/tex]
We know that [tex]\(\sqrt{27}\)[/tex] is approximately 5.196, so:
[tex]\[
3 \cdot 5.196 \approx 15.588
\][/tex]
This value is far off from 4.3267487109222245, so it is not equivalent.
3. Option [tex]\(9 \sqrt[3]{3}\)[/tex]:
[tex]\[
9 \cdot \sqrt[3]{3}
\][/tex]
Again, we know that [tex]\(\sqrt[3]{3}\)[/tex] is approximately 1.4422, so:
[tex]\[
9 \cdot 1.4422 \approx 12.9798
\][/tex]
This value is also far off from 4.3267487109222245, so it is not equivalent.
4. Option [tex]\(27 \sqrt[3]{3}\)[/tex]:
[tex]\[
27 \cdot \sqrt[3]{3}
\][/tex]
And, using [tex]\(\sqrt[3]{3} \approx 1.4422\)[/tex]:
[tex]\[
27 \cdot 1.4422 \approx 38.9394
\][/tex]
This value is much larger than 4.3267487109222245, so this cannot be the equivalent expression.
Thus, the expression that is equivalent to [tex]\(81^{\frac{1}{3}}\)[/tex] is:
[tex]\[ \boxed{3 \sqrt[3]{3}} \][/tex]
