To solve the equation for [tex]\( z \)[/tex], follow these step-by-step instructions:
Given equation:
[tex]\[
4z^3 = \frac{\left( x^{\frac{1}{2}} y^{-3} z \right)^2}{y^{-5}}
\][/tex]
First, we need to simplify the right side of the equation. Start by squaring the term inside the parentheses:
[tex]\[
\left( x^{\frac{1}{2}} y^{-3} z \right)^2 = x^{\frac{1}{2} \cdot 2} y^{-3 \cdot 2} z^2
\][/tex]
This simplifies to:
[tex]\[
x y^{-6} z^2
\][/tex]
Thus, the equation now looks like this:
[tex]\[
4z^3 = \frac{x y^{-6} z^2}{y^{-5}}
\][/tex]
Next, simplify the fraction on the right-hand side. When dividing by [tex]\( y^{-5} \)[/tex], it is equivalent to multiplying by [tex]\( y^5 \)[/tex]:
[tex]\[
\frac{x y^{-6} z^2}{y^{-5}} = x y^{-6} z^2 \cdot y^5
\][/tex]
Combine the exponents of [tex]\( y \)[/tex]:
[tex]\[
y^{-6 + 5} = y^{-1}
\][/tex]
So, the equation simplifies to:
[tex]\[
4z^3 = x y^{-1} z^2
\][/tex]
Now, we need to isolate [tex]\( z \)[/tex] on one side of the equation. Start by dividing both sides by [tex]\( z^2 \)[/tex]:
[tex]\[
4z = x y^{-1}
\][/tex]
Finally, solve for [tex]\( z \)[/tex] by dividing both sides by 4:
[tex]\[
z = \frac{x y^{-1}}{4}
\][/tex]
Since [tex]\( y^{-1} \)[/tex] is equivalent to [tex]\( \frac{1}{y} \)[/tex], the final simplified expression for [tex]\( z \)[/tex] is:
[tex]\[
z = \frac{x}{4y}
\][/tex]
So the solution is:
[tex]\[
z = \frac{x}{4y}
\][/tex]
