Solve 1/2x+1/4y(4x-8y)-(x-2y)(x+y)+(-8x^5y^3):(-4x^4y^2)

$$\frac { 1 } { 2 } x + \frac { 1 } { 4 } y ( 4 x - 8 y ) - ( x - 2 y ) ( x + y ) + ( - 8 x ^ { 5 } y ^ { 3 } ) : ( - 4 x ^ { 4 } y ^ { 2 } )$$

Evaluate

$$\frac{x\left(1+8y-2x\right)}{2}$$

Solution Steps

Use the distributive property to multiply \frac{1}{4}y by 4x-8y.

$$\frac{1}{2}x+yx-2y^{2}-\left(x-2y\right)\left(x+y\right)+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

Use the distributive property to multiply x-2y by x+y and combine like terms.

$$\frac{1}{2}x+yx-2y^{2}-\left(x^{2}-xy-2y^{2}\right)+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

To find the opposite of x^{2}-xy-2y^{2}, find the opposite of each term.

$$\frac{1}{2}x+yx-2y^{2}-x^{2}+xy+2y^{2}+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

Combine yx and xy to get 2yx.

$$\frac{1}{2}x+2yx-2y^{2}-x^{2}+2y^{2}+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

Combine -2y^{2} and 2y^{2} to get 0.

$$\frac{1}{2}x+2yx-x^{2}+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

Cancel out 4y^{2}x^{4} in both numerator and denominator.

$$\frac{1}{2}x+2yx-x^{2}+\frac{-2xy}{-1}$$

Anything divided by -1 gives its opposite.

$$\frac{1}{2}x+2yx-x^{2}+2xy$$

Combine 2yx and 2xy to get 4yx.

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

Expand

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

Solution Steps

Use the distributive property to multiply \frac{1}{4}y by 4x-8y.

$$\frac{1}{2}x+yx-2y^{2}-\left(x-2y\right)\left(x+y\right)+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

Use the distributive property to multiply x-2y by x+y and combine like terms.

$$\frac{1}{2}x+yx-2y^{2}-\left(x^{2}-xy-2y^{2}\right)+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

To find the opposite of x^{2}-xy-2y^{2}, find the opposite of each term.

$$\frac{1}{2}x+yx-2y^{2}-x^{2}+xy+2y^{2}+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

Combine yx and xy to get 2yx.

$$\frac{1}{2}x+2yx-2y^{2}-x^{2}+2y^{2}+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

Combine -2y^{2} and 2y^{2} to get 0.

$$\frac{1}{2}x+2yx-x^{2}+\frac{-8x^{5}y^{3}}{-4x^{4}y^{2}}$$

Cancel out 4y^{2}x^{4} in both numerator and denominator.

$$\frac{1}{2}x+2yx-x^{2}+\frac{-2xy}{-1}$$

Anything divided by -1 gives its opposite.

$$\frac{1}{2}x+2yx-x^{2}+2xy$$

Combine 2yx and 2xy to get 4yx.

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