what is the following sum?\n2(\\sqrt3{16x^{3}y}) + 4(\\sqrt3{54x^{6}y^{5}})\n\\(4x(\\sqrt3{2y})+12x^{2}y(\\sq…

Explanation:

Step1: Simplify the first cube - root term

Simplify $\sqrt[3]{16x^{3}y}$. We can write $16x^{3}y$ as $8x^{3}\cdot2y$. Then $\sqrt[3]{16x^{3}y}=\sqrt[3]{8x^{3}\cdot2y}=2x\sqrt[3]{2y}$. So, $2\sqrt[3]{16x^{3}y}=2\times2x\sqrt[3]{2y}=4x\sqrt[3]{2y}$.

Step2: Simplify the second cube - root term

Simplify $\sqrt[3]{54x^{6}y^{5}}$. We can write $54x^{6}y^{5}$ as $27x^{6}y^{3}\cdot2y^{2}$. Then $\sqrt[3]{54x^{6}y^{5}}=\sqrt[3]{27x^{6}y^{3}\cdot2y^{2}} = 3x^{2}y\sqrt[3]{2y^{2}}$. So, $4\sqrt[3]{54x^{6}y^{5}}=4\times3x^{2}y\sqrt[3]{2y^{2}}=12x^{2}y\sqrt[3]{2y^{2}}$.

Step3: Find the sum

The sum $2\sqrt[3]{16x^{3}y}+4\sqrt[3]{54x^{6}y^{5}}=4x\sqrt[3]{2y}+12x^{2}y\sqrt[3]{2y^{2}}$.

Answer:

$4x\left(\sqrt[3]{2y}\right)+12x^{2}y\left(\sqrt[3]{2y^{2}}\right)$