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Popular Álgebra >

expandir (1/12)(t)(L+t/2+R/2)^3

Solução

expandir

(\frac{1}{12}) (t) (L + \frac{t}{2} + \frac{R}{2})^{3}

Solução

\frac{L ^{3} t}{12} + \frac{L ^{2} t ^{2}}{8} + \frac{L ^{2} tR}{8} + \frac{L t ^{3}}{16} + \frac{L t ^{2} R}{8} + \frac{L t R ^{2}}{16} + \frac{t ^{4}}{96} + \frac{t ^{3} R}{32} + \frac{t ^{2} R ^{2}}{32} + \frac{t R ^{3}}{96}

Passos da solução

(\frac{1}{12}) (t) (L + \frac{t}{2} + \frac{R}{2})^{3}

Combinar as frações usando o mínimo múltiplo comum:

\frac{t + R}{2}

= (\frac{1}{12}) (t) (\frac{t + R}{2} + L)^{3}

Remover os parênteses:

(a) = a

= \frac{1}{12} t (L + \frac{t + R}{2})^{3}

(L + \frac{t + R}{2})^{3} = L^{3} + \frac{3 L ^{2} t + 3 L ^{2} R}{2} + \frac{3 L t ^{2} + 6 L tR + 3 L R ^{2}}{4} + \frac{t ^{3} + 3 t ^{2} R + 3 t R ^{2} + R ^{3}}{8}

= \frac{1}{12} t (\frac{3 L ^{2} t + 3 L ^{2} R}{2} + \frac{3 L t ^{2} + 6 L tR + 3 L R ^{2}}{4} + \frac{t ^{3} + 3 t ^{2} R + 3 t R ^{2} + R ^{3}}{8} + L^{3})

Multiplicar frações:

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

= \frac{1 \cdot t}{12} (\frac{3 L ^{2} t + 3 L ^{2} R}{2} + \frac{3 L t ^{2} + 6 L tR + 3 L R ^{2}}{4} + \frac{t ^{3} + 3 t ^{2} R + 3 t R ^{2} + R ^{3}}{8} + L^{3})

Multiplicar:

1 \cdot t = t

= \frac{t}{12} (\frac{3 L ^{2} t + 3 L ^{2} R}{2} + \frac{3 L t ^{2} + 6 L tR + 3 L R ^{2}}{4} + \frac{t ^{3} + 3 t ^{2} R + 3 t R ^{2} + R ^{3}}{8} + L^{3})

Multiplicar frações:

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

= \frac{t ( L ^{3} + \frac{3 L ^{2} t + 3 L ^{2} R}{2} + \frac{3 L t ^{2} + 6 L tR + 3 L R ^{2}}{4} + \frac{t ^{3} + 3 t ^{2} R + 3 t R ^{2} + R ^{3}}{8} )}{12}

Simplificar

L^{3} + \frac{3 L ^{2} t + 3 L ^{2} R}{2} + \frac{3 L t ^{2} + 6 L tR + 3 L R ^{2}}{4} + \frac{t ^{3} + 3 t ^{2} R + 3 t R ^{2} + R ^{3}}{8}

em uma fração:

\frac{8 L ^{3} + 12 L ^{2} t + 12 L ^{2} R + 6 L t ^{2} + 12 L tR + 6 L R ^{2} + t ^{3} + 3 t ^{2} R + 3 t R ^{2} + R ^{3}}{8}

= \frac{\frac{t ^{3} + 6 L t ^{2} + 3 t ^{2} R + 12 L ^{2} t + 12 L tR + 3 t R ^{2} + R ^{3} + 6 L R ^{2} + 12 L ^{2} R + 8 L ^{3}}{8} t}{12}

Multiplicar

t \frac{L ^{3} \cdot 8 + 12 L ^{2} t + 12 L ^{2} R + 6 L t ^{2} + 12 L tR + 6 L R ^{2} + t ^{3} + 3 t ^{2} R + 3 t R ^{2} + R ^{3}}{8} : \frac{8 L ^{3} t + 12 L ^{2} t ^{2} + 12 L ^{2} tR + 6 L t ^{3} + 12 L t ^{2} R + 6 L t R ^{2} + t ^{4} + 3 t ^{3} R + 3 t ^{2} R ^{2} + t R ^{3}}{8}

= \frac{\frac{8 L ^{3} t + 12 L ^{2} t ^{2} + 12 L ^{2} tR + 6 L t ^{3} + 12 L t ^{2} R + 6 L t R ^{2} + t ^{4} + 3 t ^{3} R + 3 t ^{2} R ^{2} + t R ^{3}}{8}}{12}

Aplicar as propriedades das frações:

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

= \frac{8 L ^{3} t + 12 L ^{2} t ^{2} + 12 L ^{2} tR + 6 L t ^{3} + 12 L t ^{2} R + 6 L t R ^{2} + t ^{4} + 3 t ^{3} R + 3 t ^{2} R ^{2} + t R ^{3}}{8 \cdot 12}

Multiplicar os números:

8 \cdot 12 = 96

= \frac{8 L ^{3} t + 12 L ^{2} t ^{2} + 12 L ^{2} tR + 6 L t ^{3} + 12 L t ^{2} R + 6 L t R ^{2} + t ^{4} + 3 t ^{3} R + 3 t ^{2} R ^{2} + t R ^{3}}{96}

Aplicar as propriedades das frações:

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

\frac{8 L ^{3} t + 12 L ^{2} t ^{2} + 12 L ^{2} tR + 6 L t ^{3} + 12 L t ^{2} R + 6 L t R ^{2} + t ^{4} + 3 t ^{3} R + 3 t ^{2} R ^{2} + t R ^{3}}{96} = \frac{8 L ^{3} t}{96} + \frac{12 L ^{2} t ^{2}}{96} + \frac{12 L ^{2} tR}{96} + \frac{6 L t ^{3}}{96} + \frac{12 L t ^{2} R}{96} + \frac{6 L t R ^{2}}{96} + \frac{t ^{4}}{96} + \frac{3 t ^{3} R}{96} + \frac{3 t ^{2} R ^{2}}{96} + \frac{t R ^{3}}{96}

= \frac{8 L ^{3} t}{96} + \frac{12 L ^{2} t ^{2}}{96} + \frac{12 L ^{2} tR}{96} + \frac{6 L t ^{3}}{96} + \frac{12 L t ^{2} R}{96} + \frac{6 L t R ^{2}}{96} + \frac{t ^{4}}{96} + \frac{3 t ^{3} R}{96} + \frac{3 t ^{2} R ^{2}}{96} + \frac{t R ^{3}}{96}

\frac{8 L ^{3} t}{96} = \frac{L ^{3} t}{12}

\frac{12 L ^{2} t ^{2}}{96} = \frac{L ^{2} t ^{2}}{8}

\frac{12 L ^{2} tR}{96} = \frac{L ^{2} tR}{8}

\frac{6 L t ^{3}}{96} = \frac{L t ^{3}}{16}

\frac{12 L t ^{2} R}{96} = \frac{L t ^{2} R}{8}

\frac{6 L t R ^{2}}{96} = \frac{L t R ^{2}}{16}

\frac{3 t ^{3} R}{96} = \frac{t ^{3} R}{32}

\frac{3 t ^{2} R ^{2}}{96} = \frac{t ^{2} R ^{2}}{32}

= \frac{L ^{3} t}{12} + \frac{L ^{2} t ^{2}}{8} + \frac{L ^{2} tR}{8} + \frac{L t ^{3}}{16} + \frac{L t ^{2} R}{8} + \frac{L t R ^{2}}{16} + \frac{t ^{4}}{96} + \frac{t ^{3} R}{32} + \frac{t ^{2} R ^{2}}{32} + \frac{t R ^{3}}{96}

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