Let's break down the given expression step by step:
[tex]\[
\left[\frac{\left(x^2\right)^{-1} \cdot\left(x^{2^{-1}}\right) \cdot\left(x^{-2}\right)^{-1}}{\left(x^3\right)^{-1} \cdot\left(x^{3^{-1}}\right) \cdot\left(x^{-3}\right)^{-1}}\right]^6
\][/tex]
1. Simplify the Numerator:
- [tex]\(\left(x^2\right)^{-1}\)[/tex] can be simplified to [tex]\(x^{-2}\)[/tex].
- [tex]\(x^{2^{-1}}\)[/tex] is the same as [tex]\(x^{\frac{1}{2}}\)[/tex].
- [tex]\(\left(x^{-2}\right)^{-1}\)[/tex] can be simplified to [tex]\(x^2\)[/tex].
Putting it all together, the numerator becomes:
[tex]\[
x^{-2} \cdot x^{\frac{1}{2}} \cdot x^2
\][/tex]
Using the properties of exponents [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex], we add the exponents:
[tex]\[
x^{-2 + \frac{1}{2} + 2}
\][/tex]
Simplifying the exponents:
[tex]\[
-2 + 2 + \frac{1}{2} = \frac{1}{2}
\][/tex]
So the numerator simplifies to:
[tex]\[
x^{\frac{1}{2}}
\][/tex]
2. Simplify the Denominator:
- [tex]\(\left(x^3\right)^{-1}\)[/tex] can be simplified to [tex]\(x^{-3}\)[/tex].
- [tex]\(x^{3^{-1}}\)[/tex] is the same as [tex]\(x^{\frac{1}{3}}\)[/tex].
- [tex]\(\left(x^{-3}\right)^{-1}\)[/tex] can be simplified to [tex]\(x^3\)[/tex].
Putting it all together, the denominator becomes:
[tex]\[
x^{-3} \cdot x^{\frac{1}{3}} \cdot x^3
\][/tex]
Again using the properties of exponents:
[tex]\[
x^{-3 + \frac{1}{3} + 3}
\][/tex]
Simplifying the exponents:
[tex]\[
-3 + 3 + \frac{1}{3} = \frac{1}{3}
\][/tex]
So the denominator simplifies to:
[tex]\[
x^{\frac{1}{3}}
\][/tex]
3. Forming the Fraction:
Now our expression becomes:
[tex]\[
\left[\frac{x^{\frac{1}{2}}}{x^{\frac{1}{3}}}\right]^6
\][/tex]
4. Simplify the Fraction:
By subtracting the exponents in the fraction:
[tex]\[
\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}
\][/tex]
So, we have:
[tex]\[
x^{\frac{1}{6}}
\][/tex]
5. Raise the Result to the Power of 6:
[tex]\[
\left(x^{\frac{1}{6}}\right)^6
\][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
x^{\frac{1}{6} \cdot 6} = x^1 = x
\][/tex]
After simplifying the given expression step-by-step, we conclude that the expression simplifies to:
[tex]\[
\boxed{x}
\][/tex]
