Certainly! Let's solve the given equation step-by-step.
We start with the equation:
[tex]\[
4z^3 = \frac{\left(x^{\frac{1}{2}} y^{-3} z\right)^2}{y^{-5}}
\][/tex]
### Step 1: Simplify the right-hand side
First, simplify [tex]\(\left(x^{\frac{1}{2}} y^{-3} z\right)^2\)[/tex]:
[tex]\[
\left(x^{\frac{1}{2}} y^{-3} z \right)^2 = \left(x^{\frac{1}{2}}\right)^2 \left(y^{-3}\right)^2 z^2 = x y^{-6} z^2
\][/tex]
Here, we squared each term inside the parentheses.
Next, simplify the expression by dividing by [tex]\(y^{-5}\)[/tex]:
[tex]\[
\frac{x y^{-6} z^2}{y^{-5}} = x y^{-6 + 5} z^2 = x y^{-1} z^2
\][/tex]
### Step 2: Equate the simplified right-hand side to the left-hand side
Now our equation looks like this:
[tex]\[
4z^3 = x y^{-1} z^2
\][/tex]
### Step 3: Isolate [tex]\(x\)[/tex]
To isolate [tex]\(x\)[/tex], divide both sides of the equation by [tex]\(z^2\)[/tex]:
[tex]\[
4z = x y^{-1}
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
Multiply both sides of the equation by [tex]\(y\)[/tex] to solve for [tex]\(x\)[/tex]:
[tex]\[
4z y = x
\][/tex]
### Conclusion
The solution to the equation is:
[tex]\[
x = 4 z y
\][/tex]
So, the given equation [tex]\(4z^3 = \frac{\left(x^{\frac{1}{2}} y^{-3} z\right)^2}{y^{-5}}\)[/tex] simplifies to:
[tex]\[
x = 4 z y
\][/tex]
