Solve h(x)=(frac{x/3)^frac{2{3}}+sqrt{(x/3)^frac{4{3}}-4*(x/3)^2+4}}{2/3}

$$h ( x ) = \frac { ( \frac { x } { 3 } ) ^ { \frac { 2 } { 3 } } + \sqrt { ( \frac { x } { 3 } ) ^ { \frac { 4 } { 3 } } - 4 \cdot ( \frac { x } { 3 } ) ^ { 2 } + 4 } } { \frac { 2 } { 3 } }$$

Factor

$$\frac{\sqrt{36+3^{\frac{2}{3}}x^{\frac{4}{3}}-4x^{2}}+\sqrt[3]{3}x^{\frac{2}{3}}}{2}$$

Solution Steps

Consider \left(\frac{1}{3}x\right)^{\frac{2}{3}}+\frac{1}{3}\left(3\times \left(\frac{1}{3}\right)^{\frac{1}{3}}x^{\frac{4}{3}}-4x^{2}+36\right)^{\frac{1}{2}}. Factor out \frac{1}{3}.

$$\frac{3\times 3^{-\frac{2}{3}}x^{\frac{2}{3}}+\sqrt{3\times 3^{-\frac{1}{3}}x^{\frac{4}{3}}-4x^{2}+36}}{3}$$

Rewrite the complete factored expression. Simplify.

$$\frac{3\times 3^{-\frac{2}{3}}x^{\frac{2}{3}}+\sqrt{3\times 3^{-\frac{1}{3}}x^{\frac{4}{3}}-4x^{2}+36}}{2}$$

Evaluate

$$\frac{\sqrt{36+3^{\frac{2}{3}}x^{\frac{4}{3}}-4x^{2}}+\sqrt[3]{3}x^{\frac{2}{3}}}{2}$$