To determine which values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] satisfy the equation
[tex]\[
\frac{(2xy)^4}{4xy} = 4x^a y^b,
\][/tex]
we first simplify the left-hand side of the equation.
1. Simplify the expression [tex]\((2xy)^4\)[/tex]:
[tex]\[
(2xy)^4 = (2xy)(2xy)(2xy)(2xy) = 2^4 x^4 y^4 = 16x^4 y^4.
\][/tex]
2. Now, substitute this back into the fractional term on the left-hand side:
[tex]\[
\frac{(2xy)^4}{4xy} = \frac{16x^4 y^4}{4xy}.
\][/tex]
3. Simplify the fraction by dividing the numerator and the denominator by [tex]\( 4xy \)[/tex]:
[tex]\[
\frac{16x^4 y^4}{4xy} = \frac{16x^4 y^4}{4xy} = 4 \cdot \frac{x^4 y^4}{xy} = 4 \cdot x^{4-1} y^{4-1} = 4 x^3 y^3.
\][/tex]
So, the left-hand side simplifies to:
[tex]\[
4 x^3 y^3.
\][/tex]
Next, we compare the simplified left-hand side to the right-hand side of the original equation to find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
Given:
[tex]\[
4 x^3 y^3 = 4 x^a y^b.
\][/tex]
For the equation to hold for all [tex]\( x \)[/tex] and [tex]\( y \)[/tex], the exponents on both sides must be equal:
[tex]\[
a = 3 \quad \text{and} \quad b = 3.
\][/tex]
Therefore, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that satisfy the equation are:
[tex]\[
\boxed{a = 3, b = 3}.
\][/tex]
