Answer :

To express [tex]\(3125 x^{\frac{2}{5}}\)[/tex] in its simplest radical form, let's follow these steps:

1. Understanding the Expression:
[tex]\[
3125 x^{\frac{2}{5}}
\][/tex]
Here, [tex]\(x^{\frac{2}{5}}\)[/tex] indicates a root and a power, specifically the 5th root of [tex]\(x^2\)[/tex].

2. Breaking Down the Expression:
- First, recognize that [tex]\(3125\)[/tex] is equivalent to [tex]\(5^5\)[/tex]. Hence, we can express [tex]\(3125\)[/tex] as:
[tex]\[
3125 = 5^5
\][/tex]

3. Separating the Components:
- Write the entire expression as:
[tex]\[
3125 x^{\frac{2}{5}} = 5^5 \cdot x^{\frac{2}{5}}
\][/tex]

4. Simplifying using Radicals:
- The expression [tex]\(x^{\frac{2}{5}}\)[/tex] is [tex]\( (x^2)^{\frac{1}{5}} \)[/tex], which translates to the fifth root of [tex]\(x^2\)[/tex].
- Similarly, we can express [tex]\(5^5\)[/tex] with its corresponding radical form:
[tex]\[
\sqrt[5]{(5^5)^5} \Rightarrow \sqrt{244140625}
\][/tex]

5. Combining the components:
- The expression simplifies by combining the decimal part and the [tex]\(x\)[/tex] part:
[tex]\[
244140625 \sqrt{5} \sqrt{x^2}
\][/tex]

So the simplest radical form for the expression is:

[tex]\[
244140625 \sqrt{5} \sqrt{x^2}
\][/tex]

This completes the expression in its simplest radical form.

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