Let's simplify the expression step by step. We're given the expression:
[tex]\[
\sqrt[3]{3} \cdot \sqrt[3]{9}
\][/tex]
First, we recognize that [tex]\(\sqrt[3]{3}\)[/tex] represents the cube root of 3 and can be written as:
[tex]\[
3^{1/3}
\][/tex]
Similarly, [tex]\(\sqrt[3]{9}\)[/tex] represents the cube root of 9 and can be written as:
[tex]\[
9^{1/3}
\][/tex]
Next, we express 9 in terms of its prime factors. Since [tex]\(9 = 3^2\)[/tex], we have:
[tex]\[
9^{1/3} = (3^2)^{1/3}
\][/tex]
Using the properties of exponents, specifically [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify:
[tex]\[
(3^2)^{1/3} = 3^{2/3}
\][/tex]
Now, we substitute these back into the original expression:
[tex]\[
3^{1/3} \cdot 3^{2/3}
\][/tex]
Using the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we combine the exponents:
[tex]\[
3^{1/3} \cdot 3^{2/3} = 3^{(1/3 + 2/3)}
\][/tex]
Adding the exponents, we get:
[tex]\[
3^{1/3 + 2/3} = 3^1
\][/tex]
And [tex]\(3^1\)[/tex] simplifies to:
[tex]\[
3
\][/tex]
So, the simplified expression is:
[tex]\[
\mathbf{3}
\][/tex]
Thus, the simplified form of [tex]\(\sqrt[3]{3} \cdot \sqrt[3]{9}\)[/tex] is [tex]\(\boxed{3}\)[/tex].
