Let's break down the given expression and simplify it step by step to find the constants [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
The given expression is:
[tex]\[
\frac{\sqrt[3]{xy}(zy)^2}{(xz)^{-3}\sqrt{z}}
\][/tex]
First, let's work with the numerator:
[tex]\[
\sqrt[3]{xy} \cdot (zy)^2
\][/tex]
This can be rewritten as:
[tex]\[
(xy)^{1/3} \cdot (zy)^2
\][/tex]
Next, we can break this down further:
[tex]\[
(xy)^{1/3} = x^{1/3} y^{1/3}
\][/tex]
[tex]\[
(zy)^2 = z^2 y^2
\][/tex]
So the numerator becomes:
[tex]\[
x^{1/3} y^{1/3} \cdot z^2 y^2 = x^{1/3} \cdot y^{1/3 + 2} \cdot z^2 = x^{1/3} \cdot y^{7/3} \cdot z^2
\][/tex]
Now, let's work with the denominator:
[tex]\[
(xz)^{-3} \cdot \sqrt{z}
\][/tex]
This can be rewritten as:
[tex]\[
(xz)^{-3} = x^{-3} z^{-3}
\][/tex]
[tex]\[
\sqrt{z} = z^{1/2}
\][/tex]
So the denominator becomes:
[tex]\[
x^{-3} z^{-3} \cdot z^{1/2} = x^{-3} \cdot z^{-3 + 1/2} = x^{-3} \cdot z^{-5/2}
\][/tex]
Now, we can put the numerator and denominator together:
[tex]\[
\frac{x^{1/3} y^{7/3} z^2}{x^{-3} z^{-5/2}}
\][/tex]
This can be simplified by dividing the corresponding powers:
[tex]\[
x^{1/3 - (-3)} \cdot y^{7/3} \cdot z^{2 - (-5/2)} = x^{1/3 + 3} \cdot y^{7/3} \cdot z^{2 + 5/2}
\][/tex]
Simplify the exponents:
[tex]\[
x^{1/3 + 9/3} \cdot y^{7/3} \cdot z^{4/2 + 5/2} = x^{10/3} \cdot y^{7/3} \cdot z^{9/2}
\][/tex]
Finally, notice that:
[tex]\[
x^{10/3} \cdot y^{7/3} \cdot z^{9/2}
\][/tex]
We can read off the constants [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
a = 10/3, \quad b = 7/3, \quad c = 9/2
\][/tex]
Given the constants:
[tex]\[
\boxed{a = 10/3}, \boxed{b = 7/3}, \boxed{c = 9/2}
\][/tex]
