To find the product of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] for the expression [tex]\(\frac{\sqrt{36 x^4 y^4}}{6 x^{-1} y^{-1}}\)[/tex], we need to simplify the given expression step by step.
1. Simplify the square root in the numerator:
[tex]\[
\sqrt{36 x^4 y^4}
\][/tex]
Since [tex]\(\sqrt{36} = 6\)[/tex] and [tex]\(\sqrt{x^4} = x^2\)[/tex] and [tex]\(\sqrt{y^4} = y^2\)[/tex], the expression under the square root simplifies to:
[tex]\[
\sqrt{36 x^4 y^4} = 6 x^2 y^2
\][/tex]
2. Rewrite the entire fraction:
[tex]\[
\frac{6 x^2 y^2}{6 x^{-1} y^{-1}}
\][/tex]
3. Simplify the denominator:
The expression [tex]\(6 x^{-1} y^{-1}\)[/tex] can be rewritten as:
[tex]\[
6 \cdot \frac{1}{x} \cdot \frac{1}{y} = 6 \cdot \frac{1}{xy} = \frac{6}{xy}
\][/tex]
4. Construct the full simplified expression:
[tex]\[
\frac{6 x^2 y^2}{\frac{6}{xy}} = 6 x^2 y^2 \cdot \frac{xy}{6}
\][/tex]
5. Perform cancellation and multiplication:
The [tex]\( 6 \)[/tex]'s cancel out:
[tex]\[
x^2 y^2 \cdot xy = x^(2+1) y^(2+1) = x^3 y^3
\][/tex]
Thus, the simplified expression is:
[tex]\[
x^3 y^3
\][/tex]
This expression matches the form [tex]\( a x^b y^c \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 3 \)[/tex].
6. Compute the product [tex]\( a \cdot b \cdot c \)[/tex]:
[tex]\[
a \cdot b \cdot c = 1 \cdot 3 \cdot 3 = 9
\][/tex]
Therefore, the product of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] is:
[tex]\[
\boxed{9}
\][/tex]
