Sure, let's simplify the given expression step by step:
Given:
[tex]\[
\frac{\left(x^0 y^{\frac{2}{3}} z^{-2}\right)^{\frac{3}{2}}}{\left(x^2 z^{\frac{1}{2}}\right)^{-5}}
\][/tex]
Firstly, simplify the expression inside the parentheses.
The numerator inside the parentheses is:
[tex]\[
(x^0 y^{\frac{2}{3}} z^{-2})^{\frac{3}{2}}
\][/tex]
We know that [tex]\(x^0 = 1\)[/tex] (for any non-zero [tex]\( x \)[/tex]):
[tex]\[
(1 \cdot y^{\frac{2}{3}} \cdot z^{-2})^{\frac{3}{2}} = (y^{\frac{2}{3}} z^{-2})^{\frac{3}{2}}
\][/tex]
Applying the power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(y^{\frac{2}{3}})^{\frac{3}{2}} \cdot (z^{-2})^{\frac{3}{2}} = y^{\frac{2}{3} \cdot \frac{3}{2}} \cdot z^{-2 \cdot \frac{3}{2}}
\][/tex]
Simplifying the exponents:
[tex]\[
y^{( \frac{2}{3} \cdot \frac{3}{2} )} \cdot z^{ (-2 \cdot \frac{3}{2} )} = y^1 \cdot z^{-3} = y \cdot z^{-3}
\][/tex]
Next, let's simplify the denominator:
[tex]\[
(x^2 z^{\frac{1}{2}})^{-5}
\][/tex]
Applying the power rule [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
x^{2 \cdot (-5)} \cdot z^{\frac{1}{2} \cdot (-5)} = x^{-10} \cdot z^{-\frac{5}{2}}
\][/tex]
Now, rewrite the given expression with these simplified components:
[tex]\[
\frac{y \cdot z^{-3}}{x^{-10} \cdot z^{-\frac{5}{2}}}
\][/tex]
Combine the like terms.
Dividing by [tex]\( x^{-10} \)[/tex] is equivalent to multiplying by [tex]\( x^{10} \)[/tex]:
[tex]\[
y \cdot z^{-3} \cdot x^{10}
\][/tex]
For [tex]\( z \)[/tex], as the division is given in the expression, we can subtract the exponents:
[tex]\[
z^{-3 - (-\frac{5}{2})} = z^{-3 + \frac{5}{2}}
\][/tex]
Simplifying the exponent of [tex]\( z \)[/tex]:
[tex]\[
-3 + \frac{5}{2} = -3 + 2.5 = -0.5 = -\frac{1}{2}
\][/tex]
Now the expression becomes:
[tex]\[
y \cdot x^{10} \cdot z^{-\frac{1}{2}}
\][/tex]
Finally, expressing everything cleanly, we can write:
[tex]\[
x^{10} \cdot y \cdot z^{-\frac{1}{2}}
\][/tex]
Thus, the simplified expression is:
[tex]\[
x^{10} \cdot y \cdot z^{-\frac{1}{2}}
\][/tex]
