Square Root Quotient Property: Same base and different indicator

Let's use the definition of root as a power:

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

We'll apply this definition and convert the roots in the problem:

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

Now let's recall 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:

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

Next, for convenience, we'll handle 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 subtraction (in the first fraction on the left we expanded both numerator and denominator by 5, and in the second fraction we expanded both numerator and denominator by 4), in the following steps we simplified the resulting expression,

Let's return to the problem and consider the result of the subtraction operation between the fractions we just performed, we get:

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

Let's summarize the solution steps, we found that:

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

Therefore, the correct answer is answer C.