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

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

Evaluate

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

Short Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Multiply 3 and 72 to get 216.

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

Factor the expressions that are not already factored.

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

Cancel out 8\left(x^{3}+1\right)^{\frac{3}{2}} in both numerator and denominator.

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

Expand the expression.

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

Factor

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

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

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

Multiply 3 and 72 to get 216.

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

Factor the expressions that are not already factored in \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}}}.

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

Cancel out 8\left(x^{3}+1\right)^{\frac{3}{2}} in both numerator and denominator.

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

Use the distributive property to multiply 3x by -9x^{9}-18x^{6}-9x^{3}+1.

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

Consider -27x^{10}-54x^{7}-27x^{4}+3x. Factor out 3.

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

Consider -9x^{10}-18x^{7}-9x^{4}+x. Factor out x.

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

Rewrite the complete factored expression. Simplify. Polynomial -9x^{9}-18x^{6}-9x^{3}+1 is not factored since it does not have any rational roots.

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