Solve {[x^{6}+1-(x^{3}+1)^2(x^3-1)^2]:(1/2x^2)}:(-2/3x)

$$\{ [ x ^ { 6 } + 1 - ( x ^ { 3 } + 1 ) ^ { 2 } ( x ^ { 3 } - 1 ) ^ { 2 } ] : ( \frac { 1 } { 2 } x ^ { 2 } ) \} : ( - \frac { 2 } { 3 } x )$$

Evaluate

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

Short Solution Steps

Express \frac{\frac{x^{6}+1-\left(x^{3}+1\right)^{2}\left(x^{3}-1\right)^{2}}{\frac{1}{2}x^{2}}}{-\frac{2}{3}x} as a single fraction.

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out x^{3} in both numerator and denominator.

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

Expand the expression.

$$\frac{-x^{9}+3x^{3}}{-\frac{1}{3}}$$

Divide -x^{9}+3x^{3} by -\frac{1}{3} by multiplying -x^{9}+3x^{3} by the reciprocal of -\frac{1}{3}.

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

Anything divided by -1 gives its opposite.

$$-\left(-x^{9}+3x^{3}\right)\times 3$$

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

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

To find the opposite of -3x^{9}+9x^{3}, find the opposite of each term.

$$3x^{9}-9x^{3}$$

Expand

$$3x^{9}-9x^{3}$$

Short Solution Steps

Express \frac{\frac{x^{6}+1-\left(x^{3}+1\right)^{2}\left(x^{3}-1\right)^{2}}{\frac{1}{2}x^{2}}}{-\frac{2}{3}x} as a single fraction.

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

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

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

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

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

Factor the expressions that are not already factored.

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

Cancel out x^{3} in both numerator and denominator.

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

Expand the expression.

$$\frac{-x^{9}+3x^{3}}{-\frac{1}{3}}$$

Divide -x^{9}+3x^{3} by -\frac{1}{3} by multiplying -x^{9}+3x^{3} by the reciprocal of -\frac{1}{3}.

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

Anything divided by -1 gives its opposite.

$$-\left(-x^{9}+3x^{3}\right)\times 3$$

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

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

To find the opposite of -3x^{9}+9x^{3}, find the opposite of each term.

$$3x^{9}-9x^{3}$$