The expression given is:

\[
\sqrt[4]{16m^4x} + \sqrt[4]{625m^8x} - m\sqrt[4]{x}
\]

Let's simplify each term:

1. **Simplify \(\sqrt[4]{16m^4x}\):**
   - \(16 = 2^4\)
   - \(m^4\) can be simplified as \(m\).
   - Therefore, \(\sqrt[4]{16m^4x} = \sqrt[4]{2^4 \cdot m^4 \cdot x} = 2m\sqrt[4]{x}\).

2. **Simplify \(\sqrt[4]{625m^8x}\):**
   - \(625 = 5^4\)
   - \(m^8\) can be simplified as \(m^2\) because \(\sqrt[4]{m^8} = m^2\).
   - Therefore, \(\sqrt[4]{625m^8x} = \sqrt[4]{5^4 \cdot m^8 \cdot x} = 5m^2\sqrt[4]{x}\).

3. **Simplify \(m\sqrt[4]{x}\):**
   - This term remains as it is.

Now, substituting these simplified terms back into the original expression:

\[
2m\sqrt[4]{x} + 5m^2\sqrt[4]{x} - m\sqrt[4]{x}
\]

We can factor out \(\sqrt[4]{x}\) from all terms:

\[
\sqrt[4]{x} \cdot (2m + 5m^2 - m)
\]

Finally, combine like terms inside the parentheses:

\[
\sqrt[4]{x} \cdot (5m^2 + m)
\]

So, the simplified expression is:

\[
(5m^2 + m) \sqrt[4]{x}
\]

Would you like more details on any part of the solution?

Here are five related questions:

1. How do you simplify higher-order roots for variables?
2. Can you factor quadratic expressions like \(5m^2 + m\)?
3. What are the properties of roots and exponents in algebraic expressions?
4. How would you handle an expression with different roots combined?
5. What is the significance of extracting common factors in algebraic expressions?

**Tip:** When simplifying expressions with roots, always look for common factors and try to simplify inside the root first.

Discover how to simplify the expression ^4√16m^4x + ^4√625m^8x - m^4√x step-by-step. This problem involves key algebraic concepts like roots, radicals, and factoring, making it suitable for students in Grades 9-11.

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