The expression to be simplified is:

$\sqrt[4]{\sqrt{5x}}$

Step 1: Express both radicals as fractional exponents.

• The square root $\sqrt{5x}$ can be written as $(5x)^{1/2}$.
• The fourth root $\sqrt[4]{(5x)^{1/2}}$ can then be written as $\left( (5x)^{1/2} \right)^{1/4}$.

Step 2: Multiply the exponents.

• Using the property $(a^m)^n = a^{m \cdot n}$, multiply the exponents: $\left( (5x)^{1/2} \right)^{1/4} = (5x)^{(1/2) \cdot (1/4)} = (5x)^{1/8}$

Step 3: Rewrite the expression as a single radical.

• The expression $(5x)^{1/8}$ corresponds to the eighth root of $5x$, which is written as: $\sqrt[8]{5x}$

Final Answer:

The simplified form is:

$\boxed{\sqrt[8]{5x}}$

So the correct answer is D.

Would you like more details or have any questions? Here are five related questions to consider:

1. How do you convert between radicals and fractional exponents?
2. What are the properties of exponents that help simplify nested radicals?
3. How would you solve an equation involving multiple radical expressions?
4. Can you think of a case where multiplying radicals might result in a different form?
5. What other radical simplifications involve fractional exponents?

Tip: Remember that nested radicals can always be simplified by converting them to fractional exponents!