https://file.aiquickdraw.com/mathbot/file/1726484053460mait6sf2.jpg

The expression given in the image is:

\[
\sqrt[5]{\frac{4(2n)^3}{n^8}}
\]

Let's simplify it step by step.

### Step 1: Expand the expression inside the radical.
We can first expand \((2n)^3\):
\[
(2n)^3 = 2^3 \cdot n^3 = 8n^3
\]

Now, the expression becomes:
\[
\sqrt[5]{\frac{4 \cdot 8n^3}{n^8}} = \sqrt[5]{\frac{32n^3}{n^8}}
\]

### Step 2: Simplify the fraction.
Simplifying the powers of \(n\):
\[
\frac{n^3}{n^8} = n^{3-8} = n^{-5}
\]

So the expression becomes:
\[
\sqrt[5]{32n^{-5}}
\]

### Step 3: Simplify the radical.
We can now take the 5th root of both 32 and \(n^{-5}\):
\[
\sqrt[5]{32n^{-5}} = \sqrt[5]{32} \cdot \sqrt[5]{n^{-5}}
\]

\(\sqrt[5]{32}\) is the 5th root of 32, which is approximately \(2\) (since \(2^5 = 32\)).

\(\sqrt[5]{n^{-5}} = n^{-1}\).

Thus, the simplified expression is:
\[
2n^{-1} = \frac{2}{n}
\]

### Final Answer:
\[
\boxed{\frac{2}{n}}
\]

Would you like further explanation or have any questions about this process?

Here are 5 related questions to further explore the topic:
1. How do you simplify other roots involving fractional powers?
2. What are the properties of exponents that can help in simplifying expressions like this?
3. How do you handle expressions with negative exponents?
4. What is the relationship between radicals and exponents?
5. How would you simplify an expression with multiple variables under a radical?

**Tip**: When simplifying expressions involving exponents and radicals, remember that fractional exponents represent roots.

Simplify the expression involving a fifth root: \( \sqrt[5]{\frac{4(2n)^3}{n^8}} \)

Learn how to simplify the radical expression \( \sqrt[5]{\frac{4(2n)^3}{n^8}} \). This step-by-step solution covers key algebraic concepts, including exponents and roots, making it ideal for high school students in Grades 9-12.

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