Let's tackle the problem step-by-step by simplifying the expression [tex]\(\frac{x^{\frac{2}{3}} x^{\frac{1}{2}}}{\left(x \sqrt{x^{-3}} \sqrt[3]{x^2}\right)^6}\)[/tex] and express it in the form [tex]\(x^s\)[/tex].
1. Simplify the numerator:
[tex]\[
x^{\frac{2}{3}} \cdot x^{\frac{1}{2}} = x^{\left(\frac{2}{3} + \frac{1}{2}\right)}
\][/tex]
To add the exponents, we need a common denominator:
[tex]\[
\frac{2}{3} + \frac{1}{2} = \frac{4}{6} + \frac{3}{6} = \frac{7}{6}
\][/tex]
So, the numerator becomes:
[tex]\[
x^{\frac{7}{6}}
\][/tex]
2. Simplify the denominator step-by-step:
[tex]\[
x \sqrt{x^{-3}} \sqrt[3]{x^2}
\][/tex]
Let's express each term in the same base and exponent form:
[tex]\[
x^{1} \cdot x^{-\frac{3}{2}} \cdot x^{\frac{2}{3}}
\][/tex]
Combine the exponents in the denominator:
[tex]\[
x^{\left(1 - \frac{3}{2} + \frac{2}{3}\right)}
\][/tex]
Find a common denominator to add these exponents:
[tex]\[
1 = \frac{6}{6}, \quad -\frac{3}{2} = -\frac{9}{6}, \quad \text{and} \quad \frac{2}{3} = \frac{4}{6}
\][/tex]
Add the exponents:
[tex]\[
\frac{6}{6} - \frac{9}{6} + \frac{4}{6} = \frac{1}{6}
\][/tex]
So, the expression inside the parentheses is:
[tex]\[
x^{\frac{1}{6}}
\][/tex]
The full denominator is:
[tex]\[
\left(x^{\frac{1}{6}}\right)^6 = x^{6 \cdot \frac{1}{6}} = x^{1} = x
\][/tex]
3. Combine the simplified numerator and denominator:
[tex]\[
\frac{x^{\frac{7}{6}}}{x^{1}}
\][/tex]
Subtract the exponents:
[tex]\[
x^{\frac{7}{6} - 1} = x^{\frac{7}{6} - \frac{6}{6}} = x^{\frac{1}{6}}
\][/tex]
Thus, the given expression simplifies to:
[tex]\[
x^{\frac{1}{6}}
\][/tex]
Hence, the exponent [tex]\(s\)[/tex] in the expression [tex]\(x^s\)[/tex] is:
[tex]\[
s = \frac{1}{6}
\][/tex]
