To determine which expression is equivalent to [tex]\( 81^{\frac{1}{3}} \)[/tex], we need to evaluate each given option and compare it to the original expression:
1. Option 1: [tex]\( 3 \sqrt[3]{3} \)[/tex]
Let's simplify this expression:
[tex]\[
3 \sqrt[3]{3} = 3 \cdot 3^{\frac{1}{3}}
\][/tex]
2. Option 2: [tex]\( 3 \sqrt{3^3} \)[/tex]
Simplify the inner part first:
[tex]\[
\sqrt{3^3} = \sqrt{27}
\][/tex]
And then multiply by 3:
[tex]\[
3 \sqrt{27}
\][/tex]
3. Option 3: [tex]\( 9 \sqrt[3]{3} \)[/tex]
Let's simplify this expression:
[tex]\[
9 \sqrt[3]{3} = 9 \cdot 3^{\frac{1}{3}}
\][/tex]
4. Option 4: [tex]\( 27 \sqrt[3]{3} \)[/tex]
Let's simplify this expression:
[tex]\[
27 \sqrt[3]{3} = 27 \cdot 3^{\frac{1}{3}}
\][/tex]
Now, we need to compare these values to [tex]\( 81^{\frac{1}{3}} \)[/tex].
Evaluate the original expression:
[tex]\[
81^{\frac{1}{3}} = (3^4)^{\frac{1}{3}} = 3^{\frac{4}{3}} = 3^{1 + \frac{1}{3}} = 3 \cdot 3^{\frac{1}{3}}
\][/tex]
From our simplified evaluations, we see that:
[tex]\[
81^{\frac{1}{3}} = 3 \cdot 3^{\frac{1}{3}}
\][/tex]
This matches exactly with Option 1:
[tex]\[
3 \sqrt[3]{3}
\][/tex]
Therefore, the expression equivalent to [tex]\( 81^{\frac{1}{3}} \)[/tex] is [tex]\( 3 \sqrt[3]{3} \)[/tex].
