Solve (x^{2}-1/2x)-[3x+1/2x^{3}-(1/2x^{2}+7x+x^3)+(-2x^3)^2:(2x)^3]

$$( x ^ { 2 } - \frac { 1 } { 2 } x ) - [ 3 x + \frac { 1 } { 2 } x ^ { 3 } - ( \frac { 1 } { 2 } x ^ { 2 } + 7 x + x ^ { 3 } ) + ( - 2 x ^ { 3 } ) ^ { 2 } : ( 2 x ) ^ { 3 } ]$$

Evaluate

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

Solution Steps

To find the opposite of \frac{1}{2}x^{2}+7x+x^{3}, find the opposite of each term.

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

Combine 3x and -7x to get -4x.

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

Combine \frac{1}{2}x^{3} and -x^{3} to get -\frac{1}{2}x^{3}.

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

Expand \left(-2x^{3}\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+\frac{\left(-2\right)^{2}x^{6}}{\left(2x\right)^{3}}\right)$$

Calculate -2 to the power of 2 and get 4.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+\frac{4x^{6}}{\left(2x\right)^{3}}\right)$$

Expand \left(2x\right)^{3}.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+\frac{4x^{6}}{2^{3}x^{3}}\right)$$

Calculate 2 to the power of 3 and get 8.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+\frac{4x^{6}}{8x^{3}}\right)$$

Cancel out 4x^{3} in both numerator and denominator.

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

Combine -\frac{1}{2}x^{3} and \frac{x^{3}}{2} to get 0.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{2}\right)$$

To find the opposite of -4x-\frac{1}{2}x^{2}, find the opposite of each term.

$$x^{2}-\frac{1}{2}x+4x+\frac{1}{2}x^{2}$$

Combine -\frac{1}{2}x and 4x to get \frac{7}{2}x.

$$x^{2}+\frac{7}{2}x+\frac{1}{2}x^{2}$$

Combine x^{2} and \frac{1}{2}x^{2} to get \frac{3}{2}x^{2}.

$$\frac{3}{2}x^{2}+\frac{7}{2}x$$

Expand

$$\frac{3x^{2}+7x}{2}$$

Solution Steps

To find the opposite of \frac{1}{2}x^{2}+7x+x^{3}, find the opposite of each term.

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

Combine 3x and -7x to get -4x.

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

Combine \frac{1}{2}x^{3} and -x^{3} to get -\frac{1}{2}x^{3}.

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

Expand \left(-2x^{3}\right)^{2}.

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

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+\frac{\left(-2\right)^{2}x^{6}}{\left(2x\right)^{3}}\right)$$

Calculate -2 to the power of 2 and get 4.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+\frac{4x^{6}}{\left(2x\right)^{3}}\right)$$

Expand \left(2x\right)^{3}.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+\frac{4x^{6}}{2^{3}x^{3}}\right)$$

Calculate 2 to the power of 3 and get 8.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{3}-\frac{1}{2}x^{2}+\frac{4x^{6}}{8x^{3}}\right)$$

Cancel out 4x^{3} in both numerator and denominator.

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

Combine -\frac{1}{2}x^{3} and \frac{x^{3}}{2} to get 0.

$$x^{2}-\frac{1}{2}x-\left(-4x-\frac{1}{2}x^{2}\right)$$

To find the opposite of -4x-\frac{1}{2}x^{2}, find the opposite of each term.

$$x^{2}-\frac{1}{2}x+4x+\frac{1}{2}x^{2}$$

Combine -\frac{1}{2}x and 4x to get \frac{7}{2}x.

$$x^{2}+\frac{7}{2}x+\frac{1}{2}x^{2}$$

Combine x^{2} and \frac{1}{2}x^{2} to get \frac{3}{2}x^{2}.

$$\frac{3}{2}x^{2}+\frac{7}{2}x$$