Sure! Let's simplify the given expression step-by-step.
Given expression:
[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]
### Step 1: Simplify the Numerator
Start with the numerator:
[tex]\[
\left(x^0 y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}
\][/tex]
The properties of exponents tell us that we can distribute the [tex]\(\frac{3}{2}\)[/tex] exponent to each factor inside the parenthesis:
[tex]\[
\left(x^0\right)^{\frac{3}{2}} \left(y^{\frac{2}{3}}\right)^{\frac{3}{2}} \left(z^{-2}\right)^{\frac{3}{2}}
\][/tex]
Simplify each term separately:
- [tex]\(\left(x^0\right)^{\frac{3}{2}} = x^{0 \cdot \frac{3}{2}} = x^0 = 1\)[/tex]
- [tex]\(\left(y^{\frac{2}{3}}\right)^{\frac{3}{2}} = y^{\frac{2}{3} \cdot \frac{3}{2}} = y^1 = y\)[/tex]
- [tex]\(\left(z^{-2}\right)^{\frac{3}{2}} = z^{-2 \cdot \frac{3}{2}} = z^{-3}\)[/tex]
So the numerator simplifies to:
[tex]\[
1 \cdot y \cdot z^{-3} = y z^{-3}
\][/tex]
### Step 2: Simplify the Denominator
Now, let's consider the denominator:
[tex]\[
\left(x^2 z^{\frac{1}{2}}\right)^{-0}
\][/tex]
Raising any expression to the power of zero results in 1 (as long as the base isn't zero):
[tex]\[
\left(x^2 z^{\frac{1}{2}}\right)^{-0} = 1
\][/tex]
### Step 3: Combine the Result
Now we have:
[tex]\[
\frac{y z^{-3}}{1}
\][/tex]
Which simplifies to just:
[tex]\[
y z^{-3}
\][/tex]
To write it in a single expression:
[tex]\[
y z^{-3} = \frac{y}{z^3}
\][/tex]
Thus, the simplified form of the original expression is:
[tex]\[
\boxed{\frac{y}{z^3}}
\][/tex]
This is the final simplified expression.
