To find the area of a circle where the radius is given in a specific form, follow these steps:
1. Identify the formula for the area of a circle: The area of a circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle.
2. Substitute the given radius into the formula: In this problem, the radius \( r \) is given as \( 2xy^2 \).
3. Substitute \( 2xy^2 \) for \( r \) in the area formula:
[tex]\[
A = \pi (2xy^2)^2
\][/tex]
4. Simplify the expression inside the parentheses first:
[tex]\[
(2xy^2)^2 = (2xy^2) \times (2xy^2)
\][/tex]
5. Use the properties of exponents to expand this multiplication:
[tex]\[
(2xy^2) \times (2xy^2) = 2^2 \times x^2 \times (y^2)^2
\][/tex]
6. Calculate the powers:
[tex]\[
2^2 = 4 \quad \text{and} \quad (y^2)^2 = y^{2 \times 2} = y^4
\][/tex]
Hence, it becomes:
[tex]\[
(2xy^2)^2 = 4x^2y^4
\][/tex]
7. Substitute this result back into the area formula:
[tex]\[
A = \pi \times 4x^2y^4
\][/tex]
8. Multiply the constants and keep the \( \pi \) factor:
[tex]\[
A = 4\pi x^2 y^4
\][/tex]
Therefore, the area of the circle with radius \( 2xy^2 \) expressed as a monomial is:
[tex]\[
A = 4\pi x^2 y^4
\][/tex]
