To solve the problem, we need to simplify the expression [tex]\(\frac{2 x^{-\frac{1}{4}} y^2}{\sqrt[4]{81 x^5 y^2}}\)[/tex] and rewrite it in the form [tex]\(a x^b y^c\)[/tex]. We will then find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] and calculate their product.
1. Rewrite the expression:
[tex]\[
\frac{2 x^{-\frac{1}{4}} y^2}{\sqrt[4]{81 x^5 y^2}}
\][/tex]
2. Simplify the denominator using properties of exponents and roots:
[tex]\[
\sqrt[4]{81 x^5 y^2} = (81 x^5 y^2)^{\frac{1}{4}}
\][/tex]
Since [tex]\(81 = 3^4\)[/tex], we have:
[tex]\[
(81 x^5 y^2)^{\frac{1}{4}} = 3 (x^5)^{\frac{1}{4}} (y^2)^{\frac{1}{4}} = 3 x^{\frac{5}{4}} y^{\frac{1}{2}}
\][/tex]
3. Simplify the entire expression by dividing the numerator by the simplified denominator:
[tex]\[
\frac{2 x^{-\frac{1}{4}} y^2}{3 x^{\frac{5}{4}} y^{\frac{1}{2}}}
\][/tex]
4. Combine the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[
= \frac{2}{3} \cdot \frac{x^{-\frac{1}{4}}}{x^{\frac{5}{4}}} \cdot \frac{y^2}{y^{\frac{1}{2}}}
\][/tex]
For the [tex]\(x\)[/tex] terms:
[tex]\[
x^{-\frac{1}{4}} \cdot x^{-\frac{5}{4}} = x^{-\frac{1}{4} - \frac{5}{4}} = x^{-\frac{6}{4}} = x^{-\frac{3}{2}}
\][/tex]
For the [tex]\(y\)[/tex] terms:
[tex]\[
y^2 \cdot y^{-\frac{1}{2}} = y^{2 - \frac{1}{2}} = y^{\frac{3}{2}}
\][/tex]
5. Combine all parts:
[tex]\[
\frac{2}{3} \cdot x^{-\frac{3}{2}} \cdot y^{\frac{3}{2}}
\][/tex]
Therefore, the expression has been simplified to:
[tex]\[
\frac{2}{3} \cdot x^{-\frac{3}{2}} \cdot y^{\frac{3}{2}}
\][/tex]
So, we have [tex]\(a = \frac{2}{3}\)[/tex], [tex]\(b = -\frac{3}{2}\)[/tex], and [tex]\(c = \frac{3}{2}\)[/tex].
6. Calculate the product [tex]\(a \cdot b \cdot c\)[/tex]:
[tex]\[
\left(\frac{2}{3}\right) \cdot \left(-\frac{3}{2}\right) \cdot \left(\frac{3}{2}\right)
\][/tex]
Multiplying these together:
[tex]\[
\left(\frac{2}{3} \cdot \frac{3}{2}\right) = 1 \quad \text{and} \quad 1 \cdot \left(\frac{3}{2}\right) = \left(\frac{3}{2}\right)
\][/tex]
[tex]\[
\left(\frac{3}{2}\right) \cdot \left(-1\right) = 0
\][/tex]
Thus, the product of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] is:
[tex]\[
0
\][/tex]
Therefore, the product is [tex]\(\boxed{0}\)[/tex].
