- Simplify the first term \(\sqrt{x^2 y^3}\) to \(xy\sqrt{y}\).
- Simplify the second term \(2 \sqrt{x^3 y^4}\) to \(2xy^2\sqrt{x}\).
- Combine the terms: \(xy\sqrt{y} + 2xy^2\sqrt{x} + xy\sqrt{y} = 2xy\sqrt{y} + 2xy^2\sqrt{x}\).
- The correct choice is \(\boxed{2 xy \sqrt{y} + 2 xy^2 \sqrt{x}}\).
Explanation
- Analysis of the expression
We are given the expression \(\sqrt{x^2 y^3} + 2 \sqrt{x^3 y^4} + xy \sqrt{y}\) with the constraints \(x \geq 0\) and \(y \geq 0\). Our goal is to simplify this expression by applying the properties of square roots and algebraic exponents. Since both variables are non-negative, we know that \(\sqrt{x^2} = x\) and \(\sqrt{y^2} = y\).
- Simplifying each term
We simplify each term in the expression individually.
For the first term, \(\sqrt{x^2 y^3}\):
\[\sqrt{x^2 y^3} = \sqrt{x^2} \cdot \sqrt{y^2} \cdot \sqrt{y} = x \cdot y \cdot \sqrt{y} = xy\sqrt{y}\]
For the second term, \(2 \sqrt{x^3 y^4}\):
\[2 \sqrt{x^3 y^4} = 2 \cdot \sqrt{x^2} \cdot \sqrt{x} \cdot \sqrt{(y^2)^2} = 2 \cdot x \cdot \sqrt{x} \cdot y^2 = 2xy^2\sqrt{x}\]
The third term is already in the simplest form: \(xy\sqrt{y}\).
- Combining terms and matching with options
Now we combine the simplified terms:
\[S = (xy\sqrt{y}) + (2xy^2\sqrt{x}) + (xy\sqrt{y})\]
Combining the like terms \(xy\sqrt{y} + xy\sqrt{y}\), we get:
\[S = 2xy\sqrt{y} + 2xy^2\sqrt{x}\]
Comparing this result to the given options, it matches the second option.
Examples
In geometry, when calculating the area of complex shapes involving roots or variables, such as finding the surface area of a composite solid where dimensions are given as algebraic expressions, simplifying these expressions is essential. For instance, if you are determining the dimensions of a rectangle based on root expressions, simplifying the result allows for easier comparison and practical measurement estimation, ensuring that calculations remain accurate and manageable in real-world engineering or design projects.