Chapter A.1, Problem 74E

Solution

Simplify the radical expression.

  $(3x+y)\sqrt{2y}$ .

Given:

The given expression is $\sqrt{18x^{2}y}+\sqrt{2y^3}$ .

Concept Used:

• Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
• Use the product rule to rewrite the radical as the product of two radicals.
• Simplify the root of the perfect power.

Calculation:

The given expression is $\sqrt{18x^{2}y}+\sqrt{2y^3}$ .

  $\begin{array}{c}\sqrt{18x^{2}y}+\sqrt{2y^3}=\sqrt{3\cdot 3\cdot 2x^{2}y}+\sqrt{2y^{2}y}\\ =\sqrt{3^2\cdot 2x^{2}y}+\sqrt{2y^{2}y}\\ =3x\sqrt{2y}+y\sqrt{2y}\\ =(3x+y)\sqrt{2y}\end{array}$

Thus after simplifying $\sqrt{18x^{2}y}+\sqrt{2y^3}$ we get $(3x+y)\sqrt{2y}$ .