Solve (x^{2}+3)^-1/3-2/3x^2(x^2+3)^-4/3

$$( x ^ { 2 } + 3 ) ^ { - 1 / 3 } - \frac { 2 } { 3 } x ^ { 2 } ( x ^ { 2 } + 3 ) ^ { - 4 / 3 }$$

Factor

$$\frac{x^{2}+9}{3\left(x^{2}+3\right)^{\frac{4}{3}}}$$

Solution Steps

Factor out \frac{1}{3}.

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

Consider 3\left(x^{2}+3\right)^{-\frac{1}{3}}-2x^{2}\left(x^{2}+3\right)^{-\frac{4}{3}}. Factor out \frac{1}{\sqrt[3]{x^{2}+3}}.

$$\frac{3-\frac{2x^{2}}{x^{2}+3}}{\sqrt[3]{x^{2}+3}}$$

Rewrite the complete factored expression. Polynomial x^{2}+3 is not factored since it does not have any rational roots.

$$\frac{3-\frac{2x^{2}}{x^{2}+3}}{3\sqrt[3]{x^{2}+3}}$$

Evaluate

$$\frac{x^{2}+9}{3\left(x^{2}+3\right)^{\frac{4}{3}}}$$