Let's simplify the given expression:
[tex]\[
\frac{\sqrt[4]{6}}{\sqrt[5]{6}}
\][/tex]
First, we need to express the roots as exponents. Recall that [tex]\( \sqrt[n]{a} = a^{\frac{1}{n}} \)[/tex].
Therefore, we can rewrite the expression as:
[tex]\[
\frac{6^{\frac{1}{4}}}{6^{\frac{1}{5}}}
\][/tex]
When dividing powers with the same base, we subtract the exponents:
[tex]\[
6^{\frac{1}{4}} \div 6^{\frac{1}{5}} = 6^{\left( \frac{1}{4} - \frac{1}{5} \right)}
\][/tex]
Next, we need to perform the subtraction in the exponent:
[tex]\[
\frac{1}{4} - \frac{1}{5}
\][/tex]
To subtract these fractions, we first need a common denominator. The least common multiple of 4 and 5 is 20. We convert each fraction to have this common denominator:
[tex]\[
\frac{1}{4} = \frac{5}{20} \quad \text{and} \quad \frac{1}{5} = \frac{4}{20}
\][/tex]
Now we subtract the fractions:
[tex]\[
\frac{5}{20} - \frac{4}{20} = \frac{1}{20}
\][/tex]
Thus, the exponent simplifies to:
[tex]\[
6^{\left( \frac{1}{20} \right)}
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
6^{\frac{1}{20}}
\][/tex]
So, the simplified form of [tex]\(\frac{\sqrt[4]{6}}{\sqrt[5]{6}}\)[/tex] is:
[tex]\[
6^{\frac{1}{20}}
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{6^{\frac{1}{20}}}
\][/tex]
