Answer :

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].