Solve [(x^{2}+2)^{2}-4x^{2}]^{2}-2(2x^2-1)(2x^2+1)-(x^5)^2

$$[ ( x ^ { 2 } + 2 ) ^ { 2 } - 4 x ^ { 2 } ] ^ { 2 } - 2 ( 2 x ^ { 2 } - 1 ) ( 2 x ^ { 2 } + 1 ) - ( x ^ { 5 } ) ^ { 2 }$$

Evaluate

$$18+x^{8}-x^{10}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\left(\left(x^{2}+2\right)^{2}-4x^{2}\right)^{2}-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

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

$$\left(\left(x^{2}\right)^{2}+4x^{2}+4-4x^{2}\right)^{2}-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

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

$$\left(x^{4}+4x^{2}+4-4x^{2}\right)^{2}-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

Combine 4x^{2} and -4x^{2} to get 0.

$$\left(x^{4}+4\right)^{2}-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

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

$$\left(x^{4}\right)^{2}+8x^{4}+16-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$x^{8}+8x^{4}+16-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

Use the distributive property to multiply -2 by 2x^{2}-1.

$$x^{8}+8x^{4}+16+\left(-4x^{2}+2\right)\left(2x^{2}+1\right)-x^{10}$$

Use the distributive property to multiply -4x^{2}+2 by 2x^{2}+1 and combine like terms.

$$x^{8}+8x^{4}+16-8x^{4}+2-x^{10}$$

Combine 8x^{4} and -8x^{4} to get 0.

$$x^{8}+16+2-x^{10}$$

Add 16 and 2 to get 18.

$$x^{8}+18-x^{10}$$

Expand

$$18+x^{8}-x^{10}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply 5 and 2 to get 10.

$$\left(\left(x^{2}+2\right)^{2}-4x^{2}\right)^{2}-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

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

$$\left(\left(x^{2}\right)^{2}+4x^{2}+4-4x^{2}\right)^{2}-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

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

$$\left(x^{4}+4x^{2}+4-4x^{2}\right)^{2}-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

Combine 4x^{2} and -4x^{2} to get 0.

$$\left(x^{4}+4\right)^{2}-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

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

$$\left(x^{4}\right)^{2}+8x^{4}+16-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$x^{8}+8x^{4}+16-2\left(2x^{2}-1\right)\left(2x^{2}+1\right)-x^{10}$$

Use the distributive property to multiply -2 by 2x^{2}-1.

$$x^{8}+8x^{4}+16+\left(-4x^{2}+2\right)\left(2x^{2}+1\right)-x^{10}$$

Use the distributive property to multiply -4x^{2}+2 by 2x^{2}+1 and combine like terms.

$$x^{8}+8x^{4}+16-8x^{4}+2-x^{10}$$

Combine 8x^{4} and -8x^{4} to get 0.

$$x^{8}+16+2-x^{10}$$

Add 16 and 2 to get 18.

$$x^{8}+18-x^{10}$$