To simplify the given expression $\sqrt[5]{10^4 x}$ in terms of fractional exponents and write it in the form $10^a x^b$, follow these steps:

Step-by-Step Simplification

1. Express the fifth root using fractional exponents:

   The fifth root of any expression can be written as raising that expression to the power of $\frac{1}{5}$. Therefore, we can rewrite the given expression: $\sqrt[5]{10^4 x} = (10^4 x)^{\frac{1}{5}}$

2. Apply the exponent rule to distribute the fractional exponent:

   When an expression with multiple factors is raised to a power, the exponent applies to each factor individually. Thus, we distribute the $\frac{1}{5}$ exponent to both $10^4$ and $x$: $(10^4 x)^{\frac{1}{5}} = 10^{4 \cdot \frac{1}{5}} \cdot x^{\frac{1}{5}}$

3. Simplify the exponents:

   Simplify $4 \cdot \frac{1}{5}$ to $\frac{4}{5}$: $10^{\frac{4}{5}} \cdot x^{\frac{1}{5}}$

Thus, the simplified form of the expression $\sqrt[5]{10^4 x}$ is:

$10^{\frac{4}{5}} x^{\frac{1}{5}}$

In the form $10^a x^b$:

Here, $a = \frac{4}{5}$ and $b = \frac{1}{5}$. So, the expression can be written as:

$10^{a} x^{b} = 10^{\frac{4}{5}} x^{\frac{1}{5}}$

Summary

The expression $\sqrt[5]{10^4 x}$ simplifies to $10^{\frac{4}{5}} x^{\frac{1}{5}}$.

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Would you like more details or have any questions about this simplification?

Here are some related questions for further exploration:

1. What is the general rule for simplifying expressions with fractional exponents?
2. How would you simplify $\sqrt[3]{x^2 y^4}$ in terms of fractional exponents?
3. If $a = \frac{4}{5}$ and $b = \frac{1}{5}$, what are some real-world scenarios where these fractional exponents might apply?
4. How do you convert a fractional exponent back to a radical form?
5. What are some common mistakes when simplifying expressions with fractional exponents?

Tip: When dealing with roots and exponents, always remember that a root can be expressed as a fractional exponent. For example, $\sqrt[n]{a} = a^{1/n}$.