https://file.aiquickdraw.com/mathbot/file/172693046824339m3tltr.jpg

The given expression in the image is:

\[
\sqrt[5]{243u^{25}}
\]

To simplify this expression, we can handle the constants and the variable separately.

### Step 1: Simplifying the constant \(243\)
We can express \(243\) as a power of 3:
\[
243 = 3^5
\]
Thus, the fifth root of \(243\) is:
\[
\sqrt[5]{243} = \sqrt[5]{3^5} = 3
\]

### Step 2: Simplifying the variable part \(u^{25}\)
For the variable part \(u^{25}\), we apply the rule of exponents for roots:
\[
\sqrt[5]{u^{25}} = u^{\frac{25}{5}} = u^5
\]

### Final Simplified Expression:
Combining the results, the simplified form of the expression is:
\[
\sqrt[5]{243u^{25}} = 3u^5
\]

Would you like further clarification or more details on any part of this? Here are five related questions to expand your understanding:

1. How do you simplify roots with different powers, like cube or seventh roots?
2. Can you generalize the process for variables with fractional exponents?
3. How would you handle an expression if the base was not a perfect power, like \(\sqrt[5]{250u^{10}}\)?
4. How does this process change for square roots compared to fifth roots?
5. Can the same approach be applied to a higher degree polynomial with multiple terms?

**Tip:** Remember that roots can always be converted to fractional exponents, which often makes simplification easier, especially for algebraic expressions!

What is the simplified form of the expression shown in the image?

Discover how to simplify the fifth root of an expression with exponents, specifically \( \sqrt[5]{243u^{25}} \). This guide explains the step-by-step process using exponent rules and the concept of roots, suitable for high school students.

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