Sure! Given the equations [tex]\( x = a^2 b \)[/tex] and [tex]\( y = a^3 \sqrt{b} \)[/tex], we need to express [tex]\( b \)[/tex] in terms of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Let's proceed step by step.
### Step 1: Isolate [tex]\( b \)[/tex] from the first equation
Starting with the equation:
[tex]\[ x = a^2 b \][/tex]
we can solve for [tex]\( b \)[/tex] by isolating it:
[tex]\[ b = \frac{x}{a^2} \quad \text{(Equation 1)} \][/tex]
### Step 2: Substitute [tex]\( b \)[/tex] into the second equation
Next, take the second equation:
[tex]\[ y = a^3 \sqrt{b} \][/tex]
and substitute [tex]\( b = \frac{x}{a^2} \)[/tex] from Equation 1 into it:
[tex]\[ y = a^3 \sqrt{\frac{x}{a^2}} \][/tex]
### Step 3: Simplify the expression inside the square root
Simplify the term under the square root:
[tex]\[ \sqrt{\frac{x}{a^2}} = \frac{\sqrt{x}}{a} \][/tex]
Thus, the equation becomes:
[tex]\[ y = a^3 \cdot \frac{\sqrt{x}}{a} \][/tex]
[tex]\[ y = a^2 \sqrt{x} \][/tex]
### Step 4: Solve for [tex]\( a^2 \)[/tex]
Rearrange the equation to solve for [tex]\( a^2 \)[/tex]:
[tex]\[ a^2 = \frac{y}{\sqrt{x}} \quad \text{(Equation 2)} \][/tex]
### Step 5: Substitute [tex]\( a^2 \)[/tex] back into the expression for [tex]\( b \)[/tex]
Now, substitute [tex]\( a^2 = \frac{y}{\sqrt{x}} \)[/tex] from Equation 2 back into the expression for [tex]\( b \)[/tex] from Equation 1:
[tex]\[ b = \frac{x}{a^2} \][/tex]
[tex]\[ b = \frac{x}{\frac{y}{\sqrt{x}}} \][/tex]
### Step 6: Simplify the final expression
Simplify the expression:
[tex]\[ b = \frac{x \cdot \sqrt{x}}{y} \][/tex]
[tex]\[ b = \frac{x^{3/2}}{y} \][/tex]
### Final Result
Therefore, the expression for [tex]\( b \)[/tex] in terms of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] is:
[tex]\[ \boxed{b = \frac{x^{3/2}}{y}} \][/tex]
This concludes the derivation.
