Simplify and then write radical notation.
left parenthesis m Superscript 1 divided by 4 Baseline n Superscript 9 divided by 4 Baseline right parenthesis Superscript 4 divided by 5

To simplify the expression \(\left( m^{\frac{1}{4}} n^{\frac{9}{4}} \right)^{\frac{4}{5}}\) and then write it in radical notation, follow these steps:

### Step 1: Apply the power of a power rule
The power of a power rule states that \(\left( a^m \right)^n = a^{m \cdot n}\). Applying this rule to both \(m\) and \(n\) inside the parentheses:

\[
\left( m^{\frac{1}{4}} n^{\frac{9}{4}} \right)^{\frac{4}{5}} = m^{\frac{1}{4} \cdot \frac{4}{5}} n^{\frac{9}{4} \cdot \frac{4}{5}}
\]

### Step 2: Simplify the exponents
Now multiply the exponents:

\[
m^{\frac{1}{4} \cdot \frac{4}{5}} = m^{\frac{1}{5}}
\]
\[
n^{\frac{9}{4} \cdot \frac{4}{5}} = n^{\frac{36}{20}} = n^{\frac{9}{5}}
\]

### Step 3: Express in radical notation
Radical notation is another way to write fractional exponents. Specifically, \(a^{\frac{m}{n}}\) can be written as \(\sqrt[n]{a^m}\).

Thus:

\[
m^{\frac{1}{5}} = \sqrt[5]{m}
\]
\[
n^{\frac{9}{5}} = \sqrt[5]{n^9}
\]

### Final Answer:
The simplified expression in radical notation is:

\[
\sqrt[5]{m} \cdot \sqrt[5]{n^9}
\]

Would you like further details or have any questions?

Here are some related questions:
1. How would the result change if the exponents were different?
2. Can we combine the radicals into a single expression?
3. What happens if the original expression was raised to a different fractional power?
4. How can you simplify if \(m\) and \(n\) are known constants?
5. What does radical notation represent in terms of roots?

**Tip:** The power of a power rule helps you quickly simplify nested exponents by multiplying them.

Simplify and then write radical notation. (m^(1/4) n^(9/4))^(4/5)

Learn how to simplify the expression (m^(1/4) n^(9/4))^(4/5) and write it in radical notation. This problem involves exponent rules and converting fractional exponents to radicals. Detailed solution provided for Grades 9-12.

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