Let's simplify the expression:
[tex]\[
\sqrt[3]{9 x^4} \cdot \sqrt[3]{3 x^8}
\][/tex]
### Step-by-Step Solution:
#### Step 1: Express each term inside the cube root
First, we note that:
[tex]\[
\sqrt[3]{9 x^4} \cdot \sqrt[3]{3 x^8}
\][/tex]
#### Step 2: Combine the terms under a single cube root
Using the property of radicals that [tex]\(\sqrt[3]{a} \cdot \sqrt[3]{b} = \sqrt[3]{a \cdot b}\)[/tex], we can combine the expressions:
[tex]\[
\sqrt[3]{9 x^4} \cdot \sqrt[3]{3 x^8} = \sqrt[3]{(9 x^4) \cdot (3 x^8)}
\][/tex]
#### Step 3: Multiply the expressions inside the radicands
Next, we multiply the terms inside the radicands:
[tex]\[
(9 x^4) \cdot (3 x^8) = 9 \cdot 3 \cdot x^4 \cdot x^8
\][/tex]
#### Step 4: Simplify the multiplication of constants and variables
[tex]\[
9 \cdot 3 = 27
\][/tex]
[tex]\[
x^4 \cdot x^8 = x^{4 + 8} = x^{12}
\][/tex]
#### Step 5: Combine the results from previous steps
[tex]\[
(9 x^4) \cdot (3 x^8) = 27 x^{12}
\][/tex]
#### Step 6: Apply the cube root to the simplified expression
[tex]\[
\sqrt[3]{27 x^{12}}
\][/tex]
#### Step 7: Simplify the cube root
We know that:
[tex]\[
\sqrt[3]{27} = 3
\][/tex]
and:
[tex]\[
\sqrt[3]{x^{12}} = x^{12 / 3} = x^4
\][/tex]
#### Step 8: Combine the results
So, we have:
[tex]\[
\sqrt[3]{27 x^{12}} = 3 x^4
\][/tex]
### Conclusion:
The simplified product is:
[tex]\[
3 x^4
\][/tex]
Thus, the correct answer is:
[tex]\[
3 x^4
\][/tex]
