Let's simplify the given expression step by step:
[tex]\[
\frac{\left(x^0 y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}}{\left(x^2 z^{\frac{1}{2}}\right)^{-6}}
\][/tex]
### Step 1: Simplify the numerator [tex]\(\left(x^0 y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}\)[/tex]
First, recognize that [tex]\(x^0 = 1\)[/tex], which simplifies our numerator:
[tex]\[
\left(1 \cdot y^{\frac{2}{3}} \cdot z^{-2}\right)^{\frac{3}{2}} = \left(y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}
\][/tex]
Now, we'll apply the exponent [tex]\(\frac{3}{2}\)[/tex] to both [tex]\(y^{\frac{2}{3}}\)[/tex] and [tex]\(z^{-2}\)[/tex]:
[tex]\[
\left(y^{\frac{2}{3}}\right)^{\frac{3}{2}} \cdot \left(z^{-2}\right)^{\frac{3}{2}}
\][/tex]
Next, multiply the exponents using [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
y^{\frac{2}{3} \cdot \frac{3}{2}} \cdot z^{-2 \cdot \frac{3}{2}} = y^1 \cdot z^{-3}
\][/tex]
So, the numerator simplifies to:
[tex]\[
y^1 \cdot z^{-3} = y \cdot z^{-3}
\][/tex]
### Step 2: Simplify the denominator [tex]\(\left(x^2 z^{\frac{1}{2}}\right)^{-6}\)[/tex]
Apply the exponent [tex]\(-6\)[/tex] to both [tex]\(x^2\)[/tex] and [tex]\(z^{\frac{1}{2}}\)[/tex]:
[tex]\[
\left(x^2\right)^{-6} \cdot \left(z^{\frac{1}{2}}\right)^{-6}
\][/tex]
Again, multiply the exponents:
[tex]\[
x^{2 \cdot (-6)} \cdot z^{\frac{1}{2} \cdot (-6)} = x^{-12} \cdot z^{-3}
\][/tex]
So, the denominator simplifies to:
[tex]\[
x^{-12} \cdot z^{-3}
\][/tex]
### Step 3: Combine the simplified numerator and denominator
The simplified expression is then:
[tex]\[
\frac{y \cdot z^{-3}}{x^{-12} \cdot z^{-3}}
\][/tex]
You can cancel out [tex]\(z^{-3}\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{y}{x^{-12}}
\][/tex]
Since dividing by [tex]\(x^{-12}\)[/tex] is equivalent to multiplying by [tex]\(x^{12}\)[/tex], the final expression becomes:
[tex]\[
y \cdot x^{12}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{yx^{12}}
\][/tex]
