To determine the cube root of [tex]\( 216 x^9 y^{18} \)[/tex], we need to take the cube root of each factor in the expression separately.
### Step-by-Step Solution:
1. Cube Root of the Constant Term:
The constant term is [tex]\(216\)[/tex]. We need to find the cube root of [tex]\(216\)[/tex]:
[tex]\[
\sqrt[3]{216} = 6
\][/tex]
2. Cube Root of [tex]\( x^9 \)[/tex]:
For the variable [tex]\( x \)[/tex], the original exponent is [tex]\(9\)[/tex]. To find the cube root, we divide this exponent by [tex]\(3\)[/tex]:
[tex]\[
\sqrt[3]{x^9} = x^{\frac{9}{3}} = x^3
\][/tex]
3. Cube Root of [tex]\( y^{18} \)[/tex]:
For the variable [tex]\( y \)[/tex], the original exponent is [tex]\(18\)[/tex]. Similarly, to find the cube root, we divide this exponent by [tex]\(3\)[/tex]:
[tex]\[
\sqrt[3]{y^{18}} = y^{\frac{18}{3}} = y^6
\][/tex]
4. Combining all Results:
Now we combine all the individual cube roots we have calculated:
[tex]\[
\sqrt[3]{216 x^9 y^{18}} = 6 x^3 y^6
\][/tex]
### Conclusion:
The cube root of [tex]\( 216 x^9 y^{18} \)[/tex] is:
[tex]\[
6 x^3 y^6
\][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{6 x^3 y^6} \][/tex]
