Divide as indicated.

numerator 4 x to the 5 power divided by denominator 9 y to the 4 power divided by denominator numerator 2 x to the 2 power divided by denominator 9 y to the 2 power

To divide the given expression, we need to follow the order of operations, which is known as PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, and Addition and Subtraction from left to right).

Let's break down the given expression step by step:

1. We have the numerator: 4x^5. This means we have 4 multiplied by x raised to the power of 5.
2. We have the first denominator: 9y^4. This means we have 9 multiplied by y raised to the power of 4.
3. We have the second denominator: 2x^2. This means we have 2 multiplied by x raised to the power of 2.
4. Finally, we have the numerator: 9y^2. This means we have 9 multiplied by y raised to the power of 2.

Now let's divide these expressions step by step:

Step 1: Divide the numerator by the first denominator (4x^5 / 9y^4).
- Divide the coefficients: 4 / 9 = 4/9.
- Divide the variables: x^5 / y^4 = x^(5-4) = x^1 = x.
So, the first quotient is (4/9)x.

Step 2: Divide the first quotient by the second denominator ((4/9)x / 2x^2).
- Divide the coefficients: (4/9) / 2 = 4/9 * 1/2 = 4/18 = 2/9.
- Divide the variables: x / x^2 = x^(1-2) = x^(-1) = 1/x.
So, the second quotient is (2/9)(1/x) = 2/9x.

Step 3: Divide the second quotient by the second numerator ((2/9x) / (9y^2)).
- Divide the coefficients: (2/9) / 9 = 2/9 * 1/9 = 2/81.
- Divide the variables: 1/x / y^2 = (1/x)/(y^2) = (1/x)(1/y^2) = 1/(xy^2).
So, the final quotient is (2/81) / (xy^2) = 2/(81xy^2), which simplifies to 2/81xy^2.

Therefore, the division of the given expression is 2/81xy^2.