Choose the correct simplification of $\left(2 x y^2\right)^2\left(y^2\right)^3$.

A. $2 x^2 y^{10}$
B. $4 x^2 y^9$
C. $4 x^2 y^{10}$
D. $2 x^2 y^9$

Apply the power of a product rule to get ( 2 x y 2 ) 2 = 4 x 2 y 4 .
Apply the power of a power rule to get ( y 2 ) 3 = y 6 .
Multiply the simplified terms: ( 4 x 2 y 4 ) ( y 6 ) = 4 x 2 y 10 .
The correct simplification is 4 x 2 y 10 ​ .

Explanation

Understanding the problem We are given the expression ( 2 x y 2 ) 2 ( y 2 ) 3 and asked to simplify it. We will use the rules of exponents to simplify this expression.

Applying the power of a product rule First, we apply the power of a product rule to the first term: ( 2 x y 2 ) 2 = 2 2 x 2 ( y 2 ) 2 = 4 x 2 ( y 2 ) 2

Simplifying the first term Next, we apply the power of a power rule to simplify ( y 2 ) 2 : ( y 2 ) 2 = y 2 × 2 = y 4 So, the first term becomes 4 x 2 y 4 .

Simplifying the second term Now, we apply the power of a power rule to the second term: ( y 2 ) 3 = y 2 × 3 = y 6

Multiplying the terms Finally, we multiply the simplified terms: ( 4 x 2 y 4 ) ( y 6 ) = 4 x 2 y 4 + 6 = 4 x 2 y 10

Final Answer Therefore, the correct simplification is 4 x 2 y 10 .


Examples
Understanding how to simplify expressions with exponents is crucial in many areas, such as calculating the area or volume of geometric shapes. For example, if you have a square with side length 2 x y 2 , its area would be ( 2 x y 2 ) 2 = 4 x 2 y 4 . Similarly, these skills are essential in physics when dealing with quantities that scale with powers, like the energy of a photon or the gravitational force between two objects. Mastering these algebraic manipulations provides a solid foundation for more advanced problem-solving in both mathematics and science.

Answered by GinnyAnswer | 2025-07-08

The expression ( 2 x y 2 ) 2 ( y 2 ) 3 simplifies to 4 x 2 y 10 . The correct answer is option C. 4 x^2 y^{10}.
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Answered by Anonymous | 2025-07-10