Question

what is the following sum? assume ( x geq 0 ) and ( y geq 0 ).( sqrt{x^2 y^3} + 2sqrt{x^3 y^4} + xysqrt{y} ) four multiple - choice options are shown below: ( 2xysqrt{y}+2xy^2sqrt{x} ), ( 2xysqrt{xy} ), ( x^2 y^2sqrt{y}-2xy^2sqrt{x} ), ( 4xysqrt{x^5 y^8} )

Explanation:

Step1: Simplify each radical term

First, simplify \(\sqrt{x^{2}y^{3}}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) and \(\sqrt{a^{n}} = a^{\frac{n}{2}}\) (for \(a\geq0\)), we have:
\(\sqrt{x^{2}y^{3}}=\sqrt{x^{2}}\cdot\sqrt{y^{2}\cdot y}=x\cdot y\sqrt{y}=xy\sqrt{y}\) (since \(x\geq0,y\geq0\))

Next, simplify \(2\sqrt{x^{3}y^{4}}\):
\(2\sqrt{x^{3}y^{4}} = 2\sqrt{x^{2}\cdot x\cdot(y^{2})^{2}}=2\cdot x\cdot y^{2}\cdot\sqrt{x}=2xy^{2}\sqrt{x}\)

Then, the third term is \(xy\sqrt{y}\), which is already simplified.

Step2: Combine like terms

Now, we have the three terms: \(xy\sqrt{y}\), \(2xy^{2}\sqrt{x}\), and \(xy\sqrt{y}\).
Combine the like terms \(xy\sqrt{y}\) and \(xy\sqrt{y}\):
\(xy\sqrt{y}+xy\sqrt{y}=2xy\sqrt{y}\)

So the sum of the three terms is \(2xy\sqrt{y}+2xy^{2}\sqrt{x}\) (wait, no, wait. Wait, let's re - check. Wait, the original expression is \(\sqrt{x^{2}y^{3}}+2\sqrt{x^{3}y^{4}}+xy\sqrt{y}\). So substituting the simplified forms:

\(\sqrt{x^{2}y^{3}}=xy\sqrt{y}\), \(2\sqrt{x^{3}y^{4}} = 2xy^{2}\sqrt{x}\), and the third term is \(xy\sqrt{y}\). So adding them together: \(xy\sqrt{y}+2xy^{2}\sqrt{x}+xy\sqrt{y}=2xy\sqrt{y}+2xy^{2}\sqrt{x}\), which matches the first option (the right - most option in the image: \(2xy\sqrt{y}+2xy^{2}\sqrt{x}\))

Answer:

The sum is \(2xy\sqrt{y}+2xy^{2}\sqrt{x}\) (the right - most option in the given set of options)