what is the following sum?\n\\(\\sqrt3{125x^{10}y^{13}}+sqrt3{27x^{10}y^{13}}\\)\n\\(8x^{3}y^{4}(\\sqrt3{xy})…

Explanation:

Step1: Simplify cube - roots separately

We know that (\sqrt[3]{125x^{10}y^{13}}=\sqrt[3]{125}\cdot\sqrt[3]{x^{9}\cdot x}\cdot\sqrt[3]{y^{12}\cdot y}). Since (\sqrt[3]{125} = 5), (\sqrt[3]{x^{9}}=x^{3}), (\sqrt[3]{y^{12}} = y^{4}), then (\sqrt[3]{125x^{10}y^{13}}=5x^{3}y^{4}\sqrt[3]{xy}). Also, (\sqrt[3]{27x^{10}y^{13}}=\sqrt[3]{27}\cdot\sqrt[3]{x^{9}\cdot x}\cdot\sqrt[3]{y^{12}\cdot y}). Since (\sqrt[3]{27}=3), (\sqrt[3]{x^{9}} = x^{3}), (\sqrt[3]{y^{12}}=y^{4}), then (\sqrt[3]{27x^{10}y^{13}}=3x^{3}y^{4}\sqrt[3]{xy}).

Step2: Add the simplified expressions

(\sqrt[3]{125x^{10}y^{13}}+\sqrt[3]{27x^{10}y^{13}}=(5x^{3}y^{4}\sqrt[3]{xy})+(3x^{3}y^{4}\sqrt[3]{xy})=(5 + 3)x^{3}y^{4}\sqrt[3]{xy}=8x^{3}y^{4}\sqrt[3]{xy})

Answer:

(8x^{3}y^{4}(\sqrt[3]{xy}))