what is the following sum? assume x≥0 and y≥0. (sqrt{x^{2}y^{3}}+2sqrt{x^{3}y^{4}}+xysqrt{y})…

what is the following sum? assume x≥0 and y≥0. (sqrt{x^{2}y^{3}}+2sqrt{x^{3}y^{4}}+xysqrt{y}) \n(x^{2}y^{2}sqrt{y}-2xy^{2}sqrt{x}) \n(2xysqrt{y}+2xy^{2}sqrt{x}) \n(4xysqrt{x^{5}y^{8}}) \n(2xysqrt{xy})

what is the following sum? assume x≥0 and y≥0. (sqrt{x^{2}y^{3}}+2sqrt{x^{3}y^{4}}+xysqrt{y}) \n(x^{2}y^{2}sqrt{y}-2xy^{2}sqrt{x}) \n(2xysqrt{y}+2xy^{2}sqrt{x}) \n(4xysqrt{x^{5}y^{8}}) \n(2xysqrt{xy})

Answer

Explanation:

Step1: Simplify each square - root term

For $\sqrt{x^{2}y^{3}}$, using the property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ ($a = x^{2}$, $b = y^{3}$) and $\sqrt{a^{m}}=a^{\frac{m}{2}}$, we have $\sqrt{x^{2}y^{3}}=\sqrt{x^{2}}\cdot\sqrt{y^{2}\cdot y}=xy\sqrt{y}$. For $2\sqrt{x^{3}y^{4}}$, we have $2\sqrt{x^{2}\cdot x\cdot y^{4}} = 2\sqrt{x^{2}}\cdot\sqrt{y^{4}}\cdot\sqrt{x}=2xy^{2}\sqrt{x}$. The third - term is $xy\sqrt{y}$.

Step2: Combine like terms

$xy\sqrt{y}+2xy^{2}\sqrt{x}+xy\sqrt{y}=2xy\sqrt{y}+2xy^{2}\sqrt{x}$.

Answer:

B. $2xy\sqrt{y}+2xy^{2}\sqrt{x}$