Let's start with the given formula:
[tex]\[ V=\left(\frac{a x^2 y}{w-y}\right)^{1 / 3} \][/tex]
We want to make [tex]\( f \)[/tex] the subject of this formula (i.e., isolate [tex]\( \frac{a x^2 y}{w-y} \)[/tex]).
### Step 1: Cube Both Sides
First, to eliminate the cube root, we cube both sides of the equation. This gives us:
[tex]\[ V^3 = \left(\frac{a x^2 y}{w-y}\right) \][/tex]
### Step 2: Isolate the Term
Next, we need to isolate the term [tex]\( \frac{a x^2 y}{w-y} \)[/tex]. We do this by manipulating the equation:
[tex]\[ V^3 = \frac{a x^2 y}{ w - y } \][/tex]
Let's multiply both sides of the equation by [tex]\( (w - y) \)[/tex] to get rid of the denominator on the right-hand side:
[tex]\[ V^3 (w - y) = a x^2 y \][/tex]
### Step 3: Make [tex]\(\frac{a x^2 y}{w - y}\)[/tex] the Subject
Finally, to solve for [tex]\( \frac{a x^2 y}{w - y} \)[/tex], we divide both sides by [tex]\( y \)[/tex] to isolate the term:
[tex]\[ \frac{a x^2 y}{w - y} = V^3 \left(\frac{w - y}{y}\right) \][/tex]
So the isolated expression for [tex]\( \frac{a x^2 y}{w - y} \)[/tex] is:
[tex]\[ V^3 \left(\frac{w - y}{y}\right) \][/tex]
In conclusion, the expression for [tex]\( \frac{a x^2 y}{w - y} \)[/tex] is:
[tex]\[ \boxed{V^3 \frac{w - y}{y}} \][/tex]
