Solve -16/3sqrt{(4-x^{2})^{3}}+-1/7sqrt{(4-x^2)^7}+8/5sqrt{(4-x^2)^5}

$$\frac { - 16 } { 3 } \sqrt { ( 4 - x ^ { 2 } ) ^ { 3 } } + - \frac { 1 } { 7 } \sqrt { ( 4 - x ^ { 2 } ) ^ { 7 } } + \frac { 8 } { 5 } \sqrt { ( 4 - x ^ { 2 } ) ^ { 5 } }$$

Factor

$$-\frac{\left(15x^{4}+48x^{2}+128\right)\left(\left(2-x\right)\left(x+2\right)\right)^{\frac{3}{2}}}{105}$$

Solution Steps

Factor out \frac{1}{105}.

$$\frac{-560\left(4-x^{2}\right)^{\frac{3}{2}}-15\left(4-x^{2}\right)^{\frac{7}{2}}+168\left(4-x^{2}\right)^{\frac{5}{2}}}{105}$$

Consider -560\left(4-x^{2}\right)^{\frac{3}{2}}-15\left(4-x^{2}\right)^{\frac{7}{2}}+168\left(4-x^{2}\right)^{\frac{5}{2}}. Factor out \left(4-x^{2}\right)^{\frac{3}{2}}.

$$\left(4-x^{2}\right)^{\frac{3}{2}}\left(-560-15\left(4-x^{2}\right)^{2}+168\left(4-x^{2}\right)\right)$$

Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: -15x^{4}-48x^{2}-128,4-x^{2}.

$$\frac{\left(4-x^{2}\right)^{\frac{3}{2}}\left(-15x^{4}-48x^{2}-128\right)}{105}$$

Evaluate

$$-\frac{\left(4-x^{2}\right)^{\frac{3}{2}}\left(15x^{4}+48x^{2}+128\right)}{105}$$