Solve left(5x+3yright)^3+3left(5x+3yright)^2left(5x-3yright)+3left(5x+3yright)left(5x-3yright)^2+left(5x-3yright)^3

$${ \left(5x+3y \right) }^{ 3 } +3 { \left(5x+3y \right) }^{ 2 } \left( 5x-3y \right) +3 \left( 5x+3y \right) { \left(5x-3y \right) }^{ 2 } + { \left(5x-3y \right) }^{ 3 }$$

Evaluate

$$1000x^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(5x+3y\right)^{3}.

$$125x^{3}+225x^{2}y+135xy^{2}+27y^{3}+3\left(5x+3y\right)^{2}\left(5x-3y\right)+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+3y\right)^{2}.

$$125x^{3}+225x^{2}y+135xy^{2}+27y^{3}+3\left(25x^{2}+30xy+9y^{2}\right)\left(5x-3y\right)+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Use the distributive property to multiply 3 by 25x^{2}+30xy+9y^{2}.

$$125x^{3}+225x^{2}y+135xy^{2}+27y^{3}+\left(75x^{2}+90xy+27y^{2}\right)\left(5x-3y\right)+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Use the distributive property to multiply 75x^{2}+90xy+27y^{2} by 5x-3y and combine like terms.

$$125x^{3}+225x^{2}y+135xy^{2}+27y^{3}+375x^{3}+225x^{2}y-135xy^{2}-81y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Combine 125x^{3} and 375x^{3} to get 500x^{3}.

$$500x^{3}+225x^{2}y+135xy^{2}+27y^{3}+225x^{2}y-135xy^{2}-81y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Combine 225x^{2}y and 225x^{2}y to get 450x^{2}y.

$$500x^{3}+450x^{2}y+135xy^{2}+27y^{3}-135xy^{2}-81y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Combine 135xy^{2} and -135xy^{2} to get 0.

$$500x^{3}+450x^{2}y+27y^{3}-81y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Combine 27y^{3} and -81y^{3} to get -54y^{3}.

$$500x^{3}+450x^{2}y-54y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-3y\right)^{2}.

$$500x^{3}+450x^{2}y-54y^{3}+3\left(5x+3y\right)\left(25x^{2}-30xy+9y^{2}\right)+\left(5x-3y\right)^{3}$$

Use the distributive property to multiply 3 by 5x+3y.

$$500x^{3}+450x^{2}y-54y^{3}+\left(15x+9y\right)\left(25x^{2}-30xy+9y^{2}\right)+\left(5x-3y\right)^{3}$$

Use the distributive property to multiply 15x+9y by 25x^{2}-30xy+9y^{2} and combine like terms.

$$500x^{3}+450x^{2}y-54y^{3}+375x^{3}-225x^{2}y-135y^{2}x+81y^{3}+\left(5x-3y\right)^{3}$$

Combine 500x^{3} and 375x^{3} to get 875x^{3}.

$$875x^{3}+450x^{2}y-54y^{3}-225x^{2}y-135y^{2}x+81y^{3}+\left(5x-3y\right)^{3}$$

Combine 450x^{2}y and -225x^{2}y to get 225x^{2}y.

$$875x^{3}+225x^{2}y-54y^{3}-135y^{2}x+81y^{3}+\left(5x-3y\right)^{3}$$

Combine -54y^{3} and 81y^{3} to get 27y^{3}.

$$875x^{3}+225x^{2}y+27y^{3}-135y^{2}x+\left(5x-3y\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(5x-3y\right)^{3}.

$$875x^{3}+225x^{2}y+27y^{3}-135y^{2}x+125x^{3}-225x^{2}y+135xy^{2}-27y^{3}$$

Combine 875x^{3} and 125x^{3} to get 1000x^{3}.

$$1000x^{3}+225x^{2}y+27y^{3}-135y^{2}x-225x^{2}y+135xy^{2}-27y^{3}$$

Combine 225x^{2}y and -225x^{2}y to get 0.

$$1000x^{3}+27y^{3}-135y^{2}x+135xy^{2}-27y^{3}$$

Combine -135y^{2}x and 135xy^{2} to get 0.

$$1000x^{3}+27y^{3}-27y^{3}$$

Combine 27y^{3} and -27y^{3} to get 0.

$$1000x^{3}$$

Expand

$$1000x^{3}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(5x+3y\right)^{3}.

$$125x^{3}+225x^{2}y+135xy^{2}+27y^{3}+3\left(5x+3y\right)^{2}\left(5x-3y\right)+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+3y\right)^{2}.

$$125x^{3}+225x^{2}y+135xy^{2}+27y^{3}+3\left(25x^{2}+30xy+9y^{2}\right)\left(5x-3y\right)+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Use the distributive property to multiply 3 by 25x^{2}+30xy+9y^{2}.

$$125x^{3}+225x^{2}y+135xy^{2}+27y^{3}+\left(75x^{2}+90xy+27y^{2}\right)\left(5x-3y\right)+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Use the distributive property to multiply 75x^{2}+90xy+27y^{2} by 5x-3y and combine like terms.

$$125x^{3}+225x^{2}y+135xy^{2}+27y^{3}+375x^{3}+225x^{2}y-135xy^{2}-81y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Combine 125x^{3} and 375x^{3} to get 500x^{3}.

$$500x^{3}+225x^{2}y+135xy^{2}+27y^{3}+225x^{2}y-135xy^{2}-81y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Combine 225x^{2}y and 225x^{2}y to get 450x^{2}y.

$$500x^{3}+450x^{2}y+135xy^{2}+27y^{3}-135xy^{2}-81y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Combine 135xy^{2} and -135xy^{2} to get 0.

$$500x^{3}+450x^{2}y+27y^{3}-81y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Combine 27y^{3} and -81y^{3} to get -54y^{3}.

$$500x^{3}+450x^{2}y-54y^{3}+3\left(5x+3y\right)\left(5x-3y\right)^{2}+\left(5x-3y\right)^{3}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-3y\right)^{2}.

$$500x^{3}+450x^{2}y-54y^{3}+3\left(5x+3y\right)\left(25x^{2}-30xy+9y^{2}\right)+\left(5x-3y\right)^{3}$$

Use the distributive property to multiply 3 by 5x+3y.

$$500x^{3}+450x^{2}y-54y^{3}+\left(15x+9y\right)\left(25x^{2}-30xy+9y^{2}\right)+\left(5x-3y\right)^{3}$$

Use the distributive property to multiply 15x+9y by 25x^{2}-30xy+9y^{2} and combine like terms.

$$500x^{3}+450x^{2}y-54y^{3}+375x^{3}-225x^{2}y-135y^{2}x+81y^{3}+\left(5x-3y\right)^{3}$$

Combine 500x^{3} and 375x^{3} to get 875x^{3}.

$$875x^{3}+450x^{2}y-54y^{3}-225x^{2}y-135y^{2}x+81y^{3}+\left(5x-3y\right)^{3}$$

Combine 450x^{2}y and -225x^{2}y to get 225x^{2}y.

$$875x^{3}+225x^{2}y-54y^{3}-135y^{2}x+81y^{3}+\left(5x-3y\right)^{3}$$

Combine -54y^{3} and 81y^{3} to get 27y^{3}.

$$875x^{3}+225x^{2}y+27y^{3}-135y^{2}x+\left(5x-3y\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(5x-3y\right)^{3}.

$$875x^{3}+225x^{2}y+27y^{3}-135y^{2}x+125x^{3}-225x^{2}y+135xy^{2}-27y^{3}$$

Combine 875x^{3} and 125x^{3} to get 1000x^{3}.

$$1000x^{3}+225x^{2}y+27y^{3}-135y^{2}x-225x^{2}y+135xy^{2}-27y^{3}$$

Combine 225x^{2}y and -225x^{2}y to get 0.

$$1000x^{3}+27y^{3}-135y^{2}x+135xy^{2}-27y^{3}$$

Combine -135y^{2}x and 135xy^{2} to get 0.

$$1000x^{3}+27y^{3}-27y^{3}$$

Combine 27y^{3} and -27y^{3} to get 0.

$$1000x^{3}$$