Solve frac{24x(x^{3}+1)^{3/2}-3x^{2}(12x^{2}x^{3}*(x^3+1)^3/2)}{16(x^3+1)^3/2}

$$\frac { 24 x ( x ^ { 3 } + 1 ) ^ { 3 / 2 } - 3 x ^ { 2 } ( 12 x ^ { 2 } x ^ { 3 } \cdot ( x ^ { 3 } + 1 ) ^ { 3 / 2 } ) } { 16 ( x ^ { 3 } + 1 ) ^ { 3 / 2 } }$$

Evaluate

$$-\frac{9x^{7}}{4}+\frac{3x}{2}$$

Short Solution Steps

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

$$\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-3x^{4}\times 12x^{3}\left(x^{3}+1\right)^{\frac{3}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-3x^{7}\times 12\left(x^{3}+1\right)^{\frac{3}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

Multiply 3 and 12 to get 36.

$$\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-36x^{7}\left(x^{3}+1\right)^{\frac{3}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

Factor the expressions that are not already factored.

$$\frac{12x\left(x^{3}+1\right)^{\frac{3}{2}}\left(-3x^{6}+2\right)}{16\left(x^{3}+1\right)^{\frac{3}{2}}}$$

Cancel out 4\left(x^{3}+1\right)^{\frac{3}{2}} in both numerator and denominator.

$$\frac{3x\left(-3x^{6}+2\right)}{4}$$

Expand the expression.

$$\frac{-9x^{7}+6x}{4}$$

Factor

$$\frac{3x\left(2-3x^{6}\right)}{4}$$

Solution Steps

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

$$factor(\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-3x^{4}\times 12x^{3}\left(x^{3}+1\right)^{\frac{3}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

To multiply powers of the same base, add their exponents. Add 4 and 3 to get 7.

$$factor(\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-3x^{7}\times 12\left(x^{3}+1\right)^{\frac{3}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

Multiply 3 and 12 to get 36.

$$factor(\frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-36x^{7}\left(x^{3}+1\right)^{\frac{3}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

Factor the expressions that are not already factored in \frac{24x\left(x^{3}+1\right)^{\frac{3}{2}}-36x^{7}\left(x^{3}+1\right)^{\frac{3}{2}}}{16\left(x^{3}+1\right)^{\frac{3}{2}}}.

$$factor(\frac{12x\left(x^{3}+1\right)^{\frac{3}{2}}\left(-3x^{6}+2\right)}{16\left(x^{3}+1\right)^{\frac{3}{2}}})$$

Cancel out 4\left(x^{3}+1\right)^{\frac{3}{2}} in both numerator and denominator.

$$factor(\frac{3x\left(-3x^{6}+2\right)}{4})$$

Use the distributive property to multiply 3x by -3x^{6}+2.

$$factor(\frac{-9x^{7}+6x}{4})$$

Consider -9x^{7}+6x. Factor out 3.

$$3\left(-3x^{7}+2x\right)$$

Consider -3x^{7}+2x. Factor out x.

$$x\left(-3x^{6}+2\right)$$

Rewrite the complete factored expression. Simplify. Polynomial -3x^{6}+2 is not factored since it does not have any rational roots.

$$\frac{3x\left(-3x^{6}+2\right)}{4}$$