Resolver frac{24x(x^{3}+1)^{3/2}-3x^{2}(72x^{2}(x^3+1)^7/2)}{16(x^3+1)^3/2}

$$\frac { 24 x ( x ^ { 3 } + 1 ) ^ { 3 / 2 } - 3 x ^ { 2 } ( 72 x ^ { 2 } ( x ^ { 3 } + 1 ) ^ { 7 / 2 } ) } { 16 ( x ^ { 3 } + 1 ) ^ { 3 / 2 } }$$

Calcular

$$-\frac{27x^{10}}{2}-27x^{7}-\frac{27x^{4}}{2}+\frac{3x}{2}$$

Pasos de solución curta

Para multiplicar potencias da mesma base, suma os seus expoñentes. Suma 2 e 2 para obter 4.

$$\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-3x^{4}\times 72\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

Multiplica 3 e 72 para obter 216.

$$\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-216x^{4}\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

Factoriza as expresións que aínda non o están.

$$\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

Anula 8\left(x^{3}+1\right)^{\frac{3}{2}} no numerador e no denominador.

$$\frac{3x\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{2}$$

Expande a expresión.

$$\frac{-27x^{10}-54x^{7}-27x^{4}+3x}{2}$$

Factorizar

$$\frac{3x\left(1-9x^{3}-18x^{6}-9x^{9}\right)}{2}$$

Pasos da solución

Para multiplicar potencias da mesma base, suma os seus expoñentes. Suma 2 e 2 para obter 4.

$$factor(\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-3x^{4}\times 72\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

Multiplica 3 e 72 para obter 216.

$$factor(\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-216x^{4}\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

Factoriza as expresións que aínda non o están en \frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-216x^{4}\left(x^{3}+1\right)^{\frac{7}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}.

$$factor(\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

Anula 8\left(x^{3}+1\right)^{\frac{3}{2}} no numerador e no denominador.

$$factor(\frac{3x\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{2})$$

Usa a propiedade distributiva para multiplicar 3x por -9x^{9}-18x^{6}-9x^{3}+1.

$$factor(\frac{-27x^{10}-54x^{7}-27x^{4}+3x}{2})$$

Considera -27x^{10}-54x^{7}-27x^{4}+3x. Factoriza 3.

$$3\left(-9x^{10}-18x^{7}-9x^{4}+x\right)$$

Considera -9x^{10}-18x^{7}-9x^{4}+x. Factoriza x.

$$x\left(-9x^{9}-18x^{6}-9x^{3}+1\right)$$

Reescribe a expresión factorizada completa. Simplifica. O polinomio -9x^{9}-18x^{6}-9x^{3}+1 non está factorizado porque aínda que non ten ningunha raíz racional.

$$\frac{3x\left(-9x^{9}-18x^{6}-9x^{3}+1\right)}{2}$$