desarrollar (-(piL^2(90-arccos(2/L)))/(360)-4)^2

Solución

expand

\frac{π ^{2} L ^{4}}{16} - \frac{π ^{2} L ^{4} arccos ( \frac{2}{L} )}{720} + \frac{π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600} + 2 π L^{2} - \frac{π L ^{2} arccos ( \frac{2}{L} )}{45} + 16

Pasos de solución

(- \frac{π L ^{2} ( 90 - arccos ( \frac{2}{L} ) )}{360} - 4)^{2}

Expandir

π L^{2} (90 - arccos (\frac{2}{L})) : 90 π L^{2} - π L^{2} arccos (\frac{2}{L})

= (- \frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{360} - 4)^{2}

Aplicar la formula del binomio al cuadrado:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = - \frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{360}, b = 4

= (- \frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{360})^{2} - 2 (- \frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{360}) \cdot 4 + 4^{2}

Simplificar

(- \frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{360})^{2} - 2 (- \frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{360}) \cdot 4 + 4^{2} : \frac{8100 π ^{2} L ^{4} - 180 π ^{2} L ^{4} arccos ( \frac{2}{L} ) + π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600} + \frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{45} + 16

= \frac{8100 π ^{2} L ^{4} - 180 π ^{2} L ^{4} arccos ( \frac{2}{L} ) + π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600} + \frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{45} + 16

Aplicar las propiedades de las fracciones:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{8100 π ^{2} L ^{4} - 180 π ^{2} L ^{4} arccos ( \frac{2}{L} ) + π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600} = \frac{8100 π ^{2} L ^{4}}{129600} - \frac{180 π ^{2} L ^{4} arccos ( \frac{2}{L} )}{129600} + \frac{π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600}

= \frac{8100 π ^{2} L ^{4}}{129600} - \frac{180 π ^{2} L ^{4} arccos ( \frac{2}{L} )}{129600} + \frac{π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600} + \frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{45} + 16

Aplicar las propiedades de las fracciones:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{90 π L ^{2} - π L ^{2} arccos ( \frac{2}{L} )}{45} = \frac{90 π L ^{2}}{45} - \frac{π L ^{2} arccos ( \frac{2}{L} )}{45}

= \frac{8100 π ^{2} L ^{4}}{129600} - \frac{180 π ^{2} L ^{4} arccos ( \frac{2}{L} )}{129600} + \frac{π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600} + \frac{90 π L ^{2}}{45} - \frac{π L ^{2} arccos ( \frac{2}{L} )}{45} + 16

Cancelar

\frac{8100 π ^{2} L ^{4}}{129600} : \frac{π ^{2} L ^{4}}{16}

= \frac{π ^{2} L ^{4}}{16} - \frac{180 π ^{2} L ^{4} arccos ( \frac{2}{L} )}{129600} + \frac{π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600} + \frac{90 π L ^{2}}{45} - \frac{π L ^{2} arccos ( \frac{2}{L} )}{45} + 16

Cancelar

\frac{180 π ^{2} L ^{4} arccos ( \frac{2}{L} )}{129600} : \frac{π ^{2} L ^{4} arccos ( \frac{2}{L} )}{720}

= \frac{π ^{2} L ^{4}}{16} - \frac{π ^{2} L ^{4} arccos ( \frac{2}{L} )}{720} + \frac{π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600} + \frac{90 π L ^{2}}{45} - \frac{π L ^{2} arccos ( \frac{2}{L} )}{45} + 16

Dividir:

\frac{90}{45} = 2

= \frac{π ^{2} L ^{4}}{16} - \frac{π ^{2} L ^{4} arccos ( \frac{2}{L} )}{720} + \frac{π ^{2} L ^{4} arccos ^{2} ( \frac{2}{L} )}{129600} + 2 π L^{2} - \frac{π L ^{2} arccos ( \frac{2}{L} )}{45} + 16