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Let's simplify the expression provided:

\[
\sqrt[4]{10^{5x}}
\]

This is asking for the fourth root of \( 10^{5x} \). To express this in terms of fractional exponents, we use the property that:

\[
\sqrt[n]{a^m} = a^{m/n}
\]

Applying this to our expression:

\[
\sqrt[4]{10^{5x}} = 10^{\frac{5x}{4}}
\]

Now, we compare this with the form \( 10^{x^b} \). In this case, the base is \( 10 \), and we have \( x \) raised to the power of \( \frac{5}{4} \).

Thus, the simplified form of the expression is \( 10^{\frac{5}{4}x} \).

Looking at the answer choices, the correct one is:

\[
10^{\frac{5}{4}x^{\frac{1}{4}}}
\]

Would you like any more details or clarifications?

Here are some related questions to expand on this topic:
1. How do you simplify expressions with different types of roots, such as cube roots or square roots?
2. What are the rules for multiplying exponents with the same base?
3. How can you simplify fractional exponents with negative values?
4. How would you rewrite a product of two exponential terms with different bases?
5. What is the difference between rational exponents and radical expressions?

**Tip:** Always remember that fractional exponents correspond to roots, and the denominator indicates the type of root, while the numerator retains the power of the base.

Simplify the following expression in terms of fractional exponents and write it in the form 10^x^b: 4th root of 10^5x

Simplify the expression 4th root of 10^5x in terms of fractional exponents. This problem demonstrates key concepts in exponent rules and fractional exponents for high school students in Grades 9-12.

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