Let's simplify the given expression step-by-step:
[tex]\[
\frac{\left(x^0 y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}}{\left(x^2 z^{\frac{1}{2}}\right)^{0}}
\][/tex]
First, consider the denominator:
[tex]\[
\left(x^2 z^{\frac{1}{2}}\right)^0
\][/tex]
Any expression raised to the power of 0 is equal to 1, except when the base is zero. Therefore:
[tex]\[
\left(x^2 z^{\frac{1}{2}}\right)^0 = 1
\][/tex]
Now the expression simplifies to:
[tex]\[
\frac{\left(x^0 y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}}{1} = \left(x^0 y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}
\][/tex]
Next, consider the numerator:
[tex]\[
\left(x^0 y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}
\][/tex]
Break this down by applying the exponent [tex]\(\frac{3}{2}\)[/tex] to each term inside the parenthesis:
[tex]\[
(x^0)^{\frac{3}{2}} (y^{\frac{2}{3}})^{\frac{3}{2}} (z^{-2})^{\frac{3}{2}}
\][/tex]
Simplify [tex]\( (x^0)^{\frac{3}{2}} \)[/tex]:
[tex]\[
(x^0)^{\frac{3}{2}} = 1
\][/tex]
Next, simplify [tex]\( (y^{\frac{2}{3}})^{\frac{3}{2}} \)[/tex]:
[tex]\[
(y^{\frac{2}{3}})^{\frac{3}{2}} = y^{\frac{2}{3} \cdot \frac{3}{2}} = y^1 = y
\][/tex]
Then, simplify [tex]\( (z^{-2})^{\frac{3}{2}} \)[/tex]:
[tex]\[
(z^{-2})^{\frac{3}{2}} = z^{-2 \cdot \frac{3}{2}} = z^{-3}
\][/tex]
Putting all the simplified components together, we have:
[tex]\[
1 \cdot y \cdot z^{-3} = \frac{y}{z^3}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{y}{z^3}
\][/tex]
So, the detailed, step-by-step simplification of the given expression is:
[tex]\[
\frac{\left(x^0 y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}}{\left(x^2 z^{\frac{1}{2}}\right)^0} = \frac{y}{z^3}
\][/tex]
However, the final answer is given as:
[tex]\[
(y^{0.666666666666667}/z^2)^{1.5}
\][/tex]
