To simplify the expression [tex]\(\frac{\sqrt[3]{6}}{\sqrt[4]{6}}\)[/tex], we need to work with exponents.
First, let's express the roots in exponential form:
[tex]\[
\sqrt[3]{6} = 6^{\frac{1}{3}}
\][/tex]
[tex]\[
\sqrt[4]{6} = 6^{\frac{1}{4}}
\][/tex]
Now we have the expression in exponents:
[tex]\[
\frac{6^{\frac{1}{3}}}{6^{\frac{1}{4}}}
\][/tex]
When dividing exponents with the same base, we subtract the exponents:
[tex]\[
\frac{6^{\frac{1}{3}}}{6^{\frac{1}{4}}} = 6^{\frac{1}{3} - \frac{1}{4}}
\][/tex]
To proceed with the subtraction, we need a common denominator for the fractions:
[tex]\[
\frac{1}{3} = \frac{4}{12}
\][/tex]
[tex]\[
\frac{1}{4} = \frac{3}{12}
\][/tex]
Now subtract the fractions:
[tex]\[
\frac{1}{3} - \frac{1}{4} = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}
\][/tex]
So, the simplified exponent is:
[tex]\[
6^{\frac{1}{12}}
\][/tex]
Therefore, the simplified form of [tex]\(\frac{\sqrt[3]{6}}{\sqrt[4]{6}}\)[/tex] is:
[tex]\[
6^{\frac{1}{12}}
\][/tex]
Among the given options, the correct answer is:
[tex]\[
6^{\frac{1}{12}}
\][/tex]
