https://file.aiquickdraw.com/mathbot/file/1726380665908zl2zkqx5.jpg

Let's break down the expression in the image to simplify it into the form \( x^n \).

The expression inside the 4th root is:

\[
\frac{x^2 \cdot 2}{x^{3} \cdot 2}
\]

First, simplify the expression inside the root. The constant \(2\) cancels out from the numerator and the denominator, so we're left with:

\[
\frac{x^2}{x^3}
\]

Using the laws of exponents, we subtract the exponents of \(x\):

\[
x^{2-3} = x^{-1}
\]

Now, the expression inside the 4th root is \(x^{-1}\).

Next, take the 4th root of \(x^{-1}\). The 4th root of \(x^{-1}\) can be written as:

\[
x^{-\frac{1}{4}}
\]

So, the expression simplifies to:

\[
x^{-\frac{1}{4}}
\]

This is the simplified expression in the form \(x^n\), where \(n = -\frac{1}{4}\).

Would you like more details, or do you have any questions?

Here are 5 related questions you might find helpful:
1. How do we simplify expressions with rational exponents?
2. What are the properties of exponents involving roots?
3. How do we apply the laws of exponents to divide powers?
4. What is the general rule for taking roots of exponents?
5. How can you convert between radical and exponential forms?

**Tip:** When dividing powers of the same base, subtract the exponents.

Rewrite the expression in the form x^n: Simplify the expression \( \sqrt[4]{\frac{x^2 \cdot 2}{x^3 \cdot 2}} \) and express the result as \( x^n \).

Learn how to simplify expressions involving exponents and roots, such as \( \sqrt[4]{\frac{x^2 \cdot 2}{x^3 \cdot 2}} \), and express the result in the form \( x^n \). This problem covers algebraic concepts, including the laws of exponents and the conversion between radical and exponential forms. Suitable for students in Grades 9-12.

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