To simplify [tex]\(\sqrt[6]{729 x^{24}}\)[/tex], we need to address each part of the expression separately: 729 and [tex]\(x^{24}\)[/tex].
1. Simplifying 729 under the 6th root:
- We note that [tex]\(729\)[/tex] can be expressed as a power. Specifically, [tex]\(729 = 3^6\)[/tex].
- Therefore,
[tex]\[
\sqrt[6]{729} = \sqrt[6]{3^6} = (3^6)^{\frac{1}{6}} = 3
\][/tex]
2. Simplifying [tex]\(x^{24}\)[/tex] under the 6th root:
- The exponent rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] applies here.
- So,
[tex]\[
\sqrt[6]{x^{24}} = (x^{24})^{\frac{1}{6}} = x^{24 \cdot \frac{1}{6}} = x^4
\][/tex]
3. Combining the simplified terms:
- Now combine the simplified parts together:
[tex]\[
\sqrt[6]{729x^{24}} = 3 \cdot x^4
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt[6]{729 x^{24}}\)[/tex] is:
[tex]\[
\boxed{3x^4}
\][/tex]
