Solve {[(2-x^{3})^{2}(2+x^{3})^2-x^6-16]:(3/4x^4)}:(2/3x^2)

$$\{ [ ( 2 - x ^ { 3 } ) ^ { 2 } ( 2 + x ^ { 3 } ) ^ { 2 } - x ^ { 6 } - 16 ] : ( \frac { 3 } { 4 } x ^ { 4 } ) \} : ( \frac { 2 } { 3 } x ^ { 2 } )$$

Evaluate

$$2\left(x^{6}-9\right)$$

Short Solution Steps

Express \frac{\frac{\left(2-x^{3}\right)^{2}\left(2+x^{3}\right)^{2}-x^{6}-16}{\frac{3}{4}x^{4}}}{\frac{2}{3}x^{2}} as a single fraction.

$$\frac{\left(2-x^{3}\right)^{2}\left(2+x^{3}\right)^{2}-x^{6}-16}{\frac{3}{4}x^{4}\times \frac{2}{3}x^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\left(2-x^{3}\right)^{2}\left(2+x^{3}\right)^{2}-x^{6}-16}{\frac{3}{4}x^{6}\times \frac{2}{3}}$$

Multiply \frac{3}{4} and \frac{2}{3} to get \frac{1}{2}.

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

Factor the expressions that are not already factored.

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

Cancel out x^{6} in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{6}-9}{\frac{1}{2}}$$

Divide x^{6}-9 by \frac{1}{2} by multiplying x^{6}-9 by the reciprocal of \frac{1}{2}.

$$\left(x^{6}-9\right)\times 2$$

Use the distributive property to multiply x^{6}-9 by 2.

$$2x^{6}-18$$

Expand

$$2x^{6}-18$$

Short Solution Steps

Express \frac{\frac{\left(2-x^{3}\right)^{2}\left(2+x^{3}\right)^{2}-x^{6}-16}{\frac{3}{4}x^{4}}}{\frac{2}{3}x^{2}} as a single fraction.

$$\frac{\left(2-x^{3}\right)^{2}\left(2+x^{3}\right)^{2}-x^{6}-16}{\frac{3}{4}x^{4}\times \frac{2}{3}x^{2}}$$

To multiply powers of the same base, add their exponents. Add 4 and 2 to get 6.

$$\frac{\left(2-x^{3}\right)^{2}\left(2+x^{3}\right)^{2}-x^{6}-16}{\frac{3}{4}x^{6}\times \frac{2}{3}}$$

Multiply \frac{3}{4} and \frac{2}{3} to get \frac{1}{2}.

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

Factor the expressions that are not already factored.

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

Cancel out x^{6} in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{6}-9}{\frac{1}{2}}$$

Divide x^{6}-9 by \frac{1}{2} by multiplying x^{6}-9 by the reciprocal of \frac{1}{2}.

$$\left(x^{6}-9\right)\times 2$$

Use the distributive property to multiply x^{6}-9 by 2.

$$2x^{6}-18$$