To simplify the expression [tex]\(\sqrt[3]{64 a^3 x^6} \cdot y^{18}\)[/tex], we will break it down into steps:
1. Simplify the cube root:
[tex]\[
\sqrt[3]{64 a^3 x^6}
\][/tex]
Here, we need to simplify each of the components inside the cube root separately:
- [tex]\(64\)[/tex] is a perfect cube because [tex]\(64 = 4^3\)[/tex].
- [tex]\(a^3\)[/tex] is already in the form of a perfect cube.
- [tex]\(x^6\)[/tex] can be written as [tex]\((x^2)^3\)[/tex], which is a perfect cube as well.
Therefore:
[tex]\[
\sqrt[3]{64 a^3 x^6} = \sqrt[3]{4^3 \cdot a^3 \cdot (x^2)^3}
\][/tex]
2. Take the cube root of each factor:
[tex]\[
\sqrt[3]{4^3} \cdot \sqrt[3]{a^3} \cdot \sqrt[3]{(x^2)^3}
\][/tex]
Since [tex]\(\sqrt[3]{k^3} = k\)[/tex] for any [tex]\(k\)[/tex], we can simplify each term:
[tex]\[
\sqrt[3]{4^3} = 4, \quad \sqrt[3]{a^3} = a, \quad \sqrt[3]{(x^2)^3} = x^2
\][/tex]
Combining these, we get:
[tex]\[
\sqrt[3]{64 a^3 x^6} = 4 \cdot a \cdot x^2
\][/tex]
3. Combine the simplified cube root with [tex]\(y^{18}\)[/tex]:
[tex]\[
4 \cdot a \cdot x^2 \cdot y^{18}
\][/tex]
Therefore, the simplified form of the original expression is:
[tex]\[
\boxed{4 a x^2 y^{18}}
\][/tex]
