Solve (x-3y)(x+3y)(x^2+9y^2)-9(x^4-9y^4)

$$( x - 3 y ) ( x + 3 y ) ( x ^ { 2 } + 9 y ^ { 2 } ) - 9 ( x ^ { 4 } - 9 y ^ { 4 } )$$

Evaluate

$$-8x^{4}$$

Solution Steps

Use the distributive property to multiply x-3y by x+3y and combine like terms.

$$\left(x^{2}-9y^{2}\right)\left(x^{2}+9y^{2}\right)-9\left(x^{4}-9y^{4}\right)$$

Consider \left(x^{2}-9y^{2}\right)\left(x^{2}+9y^{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(x^{2}\right)^{2}-\left(9y^{2}\right)^{2}-9\left(x^{4}-9y^{4}\right)$$

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

$$x^{4}-\left(9y^{2}\right)^{2}-9\left(x^{4}-9y^{4}\right)$$

Expand \left(9y^{2}\right)^{2}.

$$x^{4}-9^{2}\left(y^{2}\right)^{2}-9\left(x^{4}-9y^{4}\right)$$

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

$$x^{4}-9^{2}y^{4}-9\left(x^{4}-9y^{4}\right)$$

Calculate 9 to the power of 2 and get 81.

$$x^{4}-81y^{4}-9\left(x^{4}-9y^{4}\right)$$

Use the distributive property to multiply -9 by x^{4}-9y^{4}.

$$x^{4}-81y^{4}-9x^{4}+81y^{4}$$

Combine x^{4} and -9x^{4} to get -8x^{4}.

$$-8x^{4}-81y^{4}+81y^{4}$$

Combine -81y^{4} and 81y^{4} to get 0.

$$-8x^{4}$$

Expand

$$-8x^{4}$$

Solution Steps

Use the distributive property to multiply x-3y by x+3y and combine like terms.

$$\left(x^{2}-9y^{2}\right)\left(x^{2}+9y^{2}\right)-9\left(x^{4}-9y^{4}\right)$$

Consider \left(x^{2}-9y^{2}\right)\left(x^{2}+9y^{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(x^{2}\right)^{2}-\left(9y^{2}\right)^{2}-9\left(x^{4}-9y^{4}\right)$$

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

$$x^{4}-\left(9y^{2}\right)^{2}-9\left(x^{4}-9y^{4}\right)$$

Expand \left(9y^{2}\right)^{2}.

$$x^{4}-9^{2}\left(y^{2}\right)^{2}-9\left(x^{4}-9y^{4}\right)$$

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

$$x^{4}-9^{2}y^{4}-9\left(x^{4}-9y^{4}\right)$$

Calculate 9 to the power of 2 and get 81.

$$x^{4}-81y^{4}-9\left(x^{4}-9y^{4}\right)$$

Use the distributive property to multiply -9 by x^{4}-9y^{4}.

$$x^{4}-81y^{4}-9x^{4}+81y^{4}$$

Combine x^{4} and -9x^{4} to get -8x^{4}.

$$-8x^{4}-81y^{4}+81y^{4}$$

Combine -81y^{4} and 81y^{4} to get 0.

$$-8x^{4}$$