To solve [tex]\(\sqrt[8]{x} \cdot \sqrt[8]{x}\)[/tex], we need to follow the properties of radicals and exponents. Let’s work through this step-by-step.
1. Understand the expression: The given expression involves multiplying two radicals with the same index [tex]\( \sqrt[8]{x} \cdot \sqrt[8]{x} \)[/tex].
2. Using the property of radicals: When you multiply two radicals with the same index, you can combine them under a single radical sign. Specifically, [tex]\( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \)[/tex].
3. Apply this property:
[tex]\[
\sqrt[8]{x} \cdot \sqrt[8]{x} = \sqrt[8]{x \cdot x}
\][/tex]
4. Simplify the expression inside the radical:
[tex]\[
\sqrt[8]{x \cdot x} = \sqrt[8]{x^2}
\][/tex]
So the expression [tex]\(\sqrt[8]{x} \cdot \sqrt[8]{x}\)[/tex], when simplified, is [tex]\(\sqrt[8]{x^2}\)[/tex].
Therefore, the answer is:
[tex]\[ \boxed{\sqrt[8]{x^2}} \][/tex]
