Question

Solve of simplify the following problem, using the properties of roots:
$\sqrt{x^2{y}^3+3x^2{y}^3}$
assuming $x$ and $y$ are positive

• A. $3xy\sqrt{y}$
• B. $2xy\sqrt{x}$
• C. $2x^{2}y\sqrt{y}$
• D. $2xy\sqrt{y}$

Answer 1

Answer: (D)

$2xy\sqrt{y}$

$2xy\sqrt{y}:$ Notice that you have two terms under the radical that both contain $x^2{y}^3$. You can add like terms together if they are under the same radical: $\sqrt{x^2{y}^3+3x^2{y}^3}=\sqrt{(1+3)x^2{y}^3}=\sqrt{4x^2{y}^3}$. Now factor out all squares and isolate them under their own radical sign:

$\sqrt{4x^2{y}^3}=\sqrt{4}\times \sqrt{x^2}\times \sqrt{y^2}\times \sqrt{y}=2xy\sqrt{y}$

(Note since $x$ and $y$ are positive $\sqrt{x^2}=x$ and $\sqrt{y^2}=y$)