Solve 2(x-2y)^-frac{1{2}}(3x+y)^frac{1{3}}-4(x-2y)^-frac{5{2}}(3x+y)^frac{7{3}}

$$2 ( x - 2 y ) ^ { - \frac { 1 } { 2 } } ( 3 x + y ) ^ { \frac { 1 } { 3 } } - 4 ( x - 2 y ) ^ { - \frac { 5 } { 2 } } ( 3 x + y ) ^ { \frac { 7 } { 3 } }$$

Factor

$$\frac{2\sqrt[3]{3x+y}\left(2y^{2}-16xy-17x^{2}\right)}{\left(x-2y\right)^{\frac{5}{2}}}$$

Solution Steps

Factor out 2.

$$2\left(\frac{\sqrt[3]{3x+y}}{\sqrt{x-2y}}-\frac{2\left(3x+y\right)^{\frac{7}{3}}}{\left(x-2y\right)^{\frac{5}{2}}}\right)$$

Consider \left(x-2y\right)^{-\frac{1}{2}}\sqrt[3]{3x+y}-2\left(x-2y\right)^{-\frac{5}{2}}\left(3x+y\right)^{\frac{7}{3}}. Factor out \frac{\sqrt[3]{3x+y}}{\sqrt{x-2y}}.

$$\frac{\sqrt[3]{3x+y}\left(1-\frac{2\left(3x+y\right)^{2}}{\left(x-2y\right)^{2}}\right)}{\sqrt{x-2y}}$$

Rewrite the complete factored expression.

$$\frac{2\sqrt[3]{3x+y}\left(1-\frac{2\left(3x+y\right)^{2}}{\left(x-2y\right)^{2}}\right)}{\sqrt{x-2y}}$$

Evaluate

$$\frac{2\sqrt[3]{3x+y}\left(-2\left(3x+y\right)^{2}+\left(x-2y\right)^{2}\right)}{\left(x-2y\right)^{\frac{5}{2}}}$$