Solve (4x^{2}-y)^{3}-[-3y(4x^{2}-y)^2-3y^2(2x-y)(2x+y)-3y^4]-2y^3

$$( 4 x ^ { 2 } - y ) ^ { 3 } - [ - 3 y ( 4 x ^ { 2 } - y ) ^ { 2 } - 3 y ^ { 2 } ( 2 x - y ) ( 2 x + y ) - 3 y ^ { 4 } ] - 2 y ^ { 3 }$$

Evaluate

$$64x^{6}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(4x^{2}-y\right)^{3}.

$$64\left(x^{2}\right)^{3}-48\left(x^{2}\right)^{2}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(4x^{2}-y\right)^{2}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

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

$$64x^{6}-48\left(x^{2}\right)^{2}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(4x^{2}-y\right)^{2}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(4x^{2}-y\right)^{2}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x^{2}-y\right)^{2}.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(16\left(x^{2}\right)^{2}-8x^{2}y+y^{2}\right)-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(16x^{4}-8x^{2}y+y^{2}\right)-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

Use the distributive property to multiply -3y by 16x^{4}-8x^{2}y+y^{2}.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

To find the opposite of -48yx^{4}+24y^{2}x^{2}-3y^{3}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}, find the opposite of each term.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}-3y^{2}\left(2x-y\right)\left(2x+y\right)\right)+3y^{4}-2y^{3}$$

Combine -y^{3} and -2y^{3} to get -3y^{3}.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}-3y^{2}\left(2x-y\right)\left(2x+y\right)\right)+3y^{4}$$

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

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}+\left(-6y^{2}x+3y^{3}\right)\left(2x+y\right)\right)+3y^{4}$$

Use the distributive property to multiply -6y^{2}x+3y^{3} by 2x+y and combine like terms.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}-12y^{2}x^{2}+3y^{4}\right)+3y^{4}$$

Combine 24y^{2}x^{2} and -12y^{2}x^{2} to get 12y^{2}x^{2}.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}-\left(-48yx^{4}+12y^{2}x^{2}-3y^{3}+3y^{4}\right)+3y^{4}$$

To find the opposite of -48yx^{4}+12y^{2}x^{2}-3y^{3}+3y^{4}, find the opposite of each term.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}+48yx^{4}-12y^{2}x^{2}+3y^{3}-3y^{4}+3y^{4}$$

Combine -48x^{4}y and 48yx^{4} to get 0.

$$64x^{6}+12x^{2}y^{2}-3y^{3}-12y^{2}x^{2}+3y^{3}-3y^{4}+3y^{4}$$

Combine 12x^{2}y^{2} and -12y^{2}x^{2} to get 0.

$$64x^{6}-3y^{3}+3y^{3}-3y^{4}+3y^{4}$$

Combine -3y^{3} and 3y^{3} to get 0.

$$64x^{6}-3y^{4}+3y^{4}$$

Combine -3y^{4} and 3y^{4} to get 0.

$$64x^{6}$$

Expand

$$64x^{6}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(4x^{2}-y\right)^{3}.

$$64\left(x^{2}\right)^{3}-48\left(x^{2}\right)^{2}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(4x^{2}-y\right)^{2}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

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

$$64x^{6}-48\left(x^{2}\right)^{2}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(4x^{2}-y\right)^{2}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(4x^{2}-y\right)^{2}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x^{2}-y\right)^{2}.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(16\left(x^{2}\right)^{2}-8x^{2}y+y^{2}\right)-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-3y\left(16x^{4}-8x^{2}y+y^{2}\right)-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

Use the distributive property to multiply -3y by 16x^{4}-8x^{2}y+y^{2}.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}\right)-2y^{3}$$

To find the opposite of -48yx^{4}+24y^{2}x^{2}-3y^{3}-3y^{2}\left(2x-y\right)\left(2x+y\right)-3y^{4}, find the opposite of each term.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}-3y^{2}\left(2x-y\right)\left(2x+y\right)\right)+3y^{4}-2y^{3}$$

Combine -y^{3} and -2y^{3} to get -3y^{3}.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}-3y^{2}\left(2x-y\right)\left(2x+y\right)\right)+3y^{4}$$

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

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}+\left(-6y^{2}x+3y^{3}\right)\left(2x+y\right)\right)+3y^{4}$$

Use the distributive property to multiply -6y^{2}x+3y^{3} by 2x+y and combine like terms.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}-\left(-48yx^{4}+24y^{2}x^{2}-3y^{3}-12y^{2}x^{2}+3y^{4}\right)+3y^{4}$$

Combine 24y^{2}x^{2} and -12y^{2}x^{2} to get 12y^{2}x^{2}.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}-\left(-48yx^{4}+12y^{2}x^{2}-3y^{3}+3y^{4}\right)+3y^{4}$$

To find the opposite of -48yx^{4}+12y^{2}x^{2}-3y^{3}+3y^{4}, find the opposite of each term.

$$64x^{6}-48x^{4}y+12x^{2}y^{2}-3y^{3}+48yx^{4}-12y^{2}x^{2}+3y^{3}-3y^{4}+3y^{4}$$

Combine -48x^{4}y and 48yx^{4} to get 0.

$$64x^{6}+12x^{2}y^{2}-3y^{3}-12y^{2}x^{2}+3y^{3}-3y^{4}+3y^{4}$$

Combine 12x^{2}y^{2} and -12y^{2}x^{2} to get 0.

$$64x^{6}-3y^{3}+3y^{3}-3y^{4}+3y^{4}$$

Combine -3y^{3} and 3y^{3} to get 0.

$$64x^{6}-3y^{4}+3y^{4}$$

Combine -3y^{4} and 3y^{4} to get 0.

$$64x^{6}$$