Use the definition of rational exponents to write each of the following with the appropriate root. Then simplify:

To rewrite expressions using rational exponents and then simplify them, let’s consider an example expression, say $a^{\frac{1}{n}}$. The rational exponent indicates the $n$-th root of $a$. Specifically, $a^{\frac{1}{n}}$ can be expressed as $\sqrt[n]{a}$.

Now let's give a structured approach to this task:

1. ### Break Down the Problem

Identify the base and the exponent in the expression that need to be rewritten using rational exponents or roots.

2. ### Relevant Concepts

Recall that:

• If $a^{\frac{m}{n}}$, this can be expressed as $\sqrt[n]{a^m}$ or $(a^m)^{\frac{1}{n}}$.

3. ### Analysis and Detail

For example, if we have $x^{\frac{3}{2}}$, you would:

• Rewrite it as $\sqrt[2]{x^3}$, which is the same as $\sqrt{x^3}$.
• You can further simplify $\sqrt{x^3}$ to $x \sqrt{x}$.

4. ### Verify and Summarize

• Double-check each simplification and ensure it adheres to the properties of exponents.
• Make sure that the transformed expression accurately represents its original value.

Final Answer

So if the original expression was $x^{\frac{3}{2}}$, the simplified version using roots is $x \sqrt{x}$.

If you have specific expressions you would like to convert and simplify, please provide them, and I can apply the above steps to assist further!