Simplify the Radical Expression: Fourth Root of 2 Divided by Fifth Root of 2

Express the definition of root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Apply this definition and proceed to convert the roots in the problem:

$\frac{\sqrt[4]{2}}{\sqrt[5]{2}}= \frac{2^{\frac{1}{4}}}{2^{\frac{1}{5}}}$

Below is the law of powers for division with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

Let's apply this law to our problem:

$\frac{2^{\frac{1}{4}}}{2^{\frac{1}{5}}}=2^{\frac{1}{4}-\frac{1}{5}}$

In order to not overly complicate our calculations proceed to solve the expression in the power numerator from the last step separately and calculate the value of the fraction:

$\frac{1}{4}-\frac{1}{5}=\frac{5\cdot1-4\cdot1}{20}=\\ \frac{5-4}{20}=\frac{1}{20}$

In the first step, we combined the two fractions into one fraction line, by expanding to the common denominator of 20 and performing the subtraction operation. (In the first fraction on the left we expanded both the numerator and denominator by 5, and in the second fraction we expanded both the numerator and denominator by 4) We then proceeded to simplify the resulting expression,

Returning once more to our problem, consider the result of the subtraction operation between the fractions that we just performed, as shown below:

$2^{\frac{1}{4}-\frac{1}{5}}=2^{\frac{1}{20}}$

Summarize the various steps of the solution:

$\frac{\sqrt[4]{2}}{\sqrt[5]{2}}=2^{\frac{1}{4}-\frac{1}{5}}=2^{\frac{1}{20}}$

Therefore, the correct answer is answer C.