Simplify the following expression in terms of fractional exponents and write it in the form 10 to the power of a x to the power of b.

fifth root of 10 to the power of 4 x end root

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, \(\sqrt[n]{a} = a^{1/n}\).

Learn how to simplify the expression \(\sqrt[5]{10^{4x}}\) using fractional exponents. This problem involves applying the rule for exponents and roots to simplify expressions effectively. Suitable for high school students and anyone interested in algebraic manipulation of exponents and roots.

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