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} )
options:
( 2xysqrt{xy} )
( 4xysqrt{x^5 y^8} )
( x^2 y^2 sqrt{y} - 2xy^2 sqrt{x} )
( 2xysqrt{y} + 2xy^2 sqrt{x} )

Explanation:

Step1: Simplify $\sqrt{x^2y^3}$

Using the property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ and $\sqrt{a^2}=a$ (for $a\geq0$), we have:
$\sqrt{x^2y^3}=\sqrt{x^2}\cdot\sqrt{y^2\cdot y}=x\cdot y\sqrt{y}=xy\sqrt{y}$

Step2: Simplify $2\sqrt{x^3y^4}$

Similarly, $\sqrt{x^3y^4}=\sqrt{x^2\cdot x}\cdot\sqrt{y^4}=x\cdot y^2\sqrt{x}$, so $2\sqrt{x^3y^4}=2xy^2\sqrt{x}$

Step3: Simplify $xy\sqrt{y}$

This term is already simplified as $xy\sqrt{y}$

Step4: Combine the terms

Now, add the three simplified terms:
$\sqrt{x^2y^3}+2\sqrt{x^3y^4}+xy\sqrt{y}=xy\sqrt{y}+2xy^2\sqrt{x}+xy\sqrt{y}$
Combine like terms (the terms with $xy\sqrt{y}$):
$xy\sqrt{y}+xy\sqrt{y}+2xy^2\sqrt{x}=2xy\sqrt{y}+2xy^2\sqrt{x}$

Answer:

$2xy\sqrt{y}+2xy^2\sqrt{x}$ (the fourth option: $2xy\sqrt{y}+2xy^2\sqrt{x}$)