To determine the simplest form of \(\sqrt[3]{x^{10}}\), we can follow a systematic process.
1. Rewrite the expression \( x^{10} \) in a simpler form:
\( x^{10} = (x^9) \cdot x \)
2. Breakdown the expression:
We need to separate \( x^{10} \) into components that can be easily managed under the cube root.
3. Use the property of exponents:
Recall that \(\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}\). Applying this to our expression, we get:
[tex]\[
\sqrt[3]{x^{10}} = \sqrt[3]{(x^9) \cdot x}
\][/tex]
4. Simplify \(\sqrt[3]{x^9}\):
Note that \( x^9 \) can be simplified under a cube root because \( (x^3)^3 = x^9 \). Therefore:
[tex]\[
\sqrt[3]{x^9} = x^3
\][/tex]
5. Combine the simplified parts:
Now we can rewrite our expression as:
[tex]\[
\sqrt[3]{x^{10}} = \sqrt[3]{(x^9) \cdot x} = \sqrt[3]{x^9} \cdot \sqrt[3]{x} = x^3 \cdot \sqrt[3]{x}
\][/tex]
Thus, the simplest form of \(\sqrt[3]{x^{10}}\) is:
[tex]\[
\boxed{x^3 \sqrt[3]{x}}
\][/tex]
