To solve the problem, we need to handle the expressions involving exponents according to the rules of exponents.
Given the expression:
[tex]\[
\frac{x^{\frac{3}{4}}}{x^{\frac{1}{6}}} \cdot x^{\frac{7}{12}}
\][/tex]
First, let's address the division. According to the exponent rules, when we divide two powers with the same base, we subtract the exponents:
[tex]\[
\frac{x^{\frac{3}{4}}}{x^{\frac{1}{6}}} = x^{\frac{3}{4} - \frac{1}{6}}
\][/tex]
To subtract the fractions [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex], we need a common denominator. The common denominator for 4 and 6 is 12. Converting each fraction:
[tex]\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
\][/tex]
[tex]\[
\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}
\][/tex]
Now, subtract the exponents:
[tex]\[
\frac{9}{12} - \frac{2}{12} = \frac{9 - 2}{12} = \frac{7}{12}
\][/tex]
So,
[tex]\[
\frac{x^{\frac{3}{4}}}{x^{\frac{1}{6}}} = x^{\frac{7}{12}}
\][/tex]
Next, we multiply this result by [tex]\(x^{\frac{7}{12}}\)[/tex]. According to the exponent rules, when we multiply two powers with the same base, we add the exponents:
[tex]\[
x^{\frac{7}{12}} \cdot x^{\frac{7}{12}} = x^{\frac{7}{12} + \frac{7}{12}} = x^{\frac{14}{12}} = x^{\frac{7}{6}}
\][/tex]
Thus, the final result is:
[tex]\[
\boxed{x^{\frac{7}{6}}}
\][/tex]
