Solve (x^2-2)^3-(x^6+8)

$$( x ^ { 2 } - 2 ) ^ { 3 } - ( x ^ { 6 } + 8 )$$

Evaluate

$$-6x^{4}+12x^{2}-16$$

Solution Steps

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

$$\left(x^{2}\right)^{3}-6\left(x^{2}\right)^{2}+12x^{2}-8-\left(x^{6}+8\right)$$

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

$$x^{6}-6\left(x^{2}\right)^{2}+12x^{2}-8-\left(x^{6}+8\right)$$

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

$$x^{6}-6x^{4}+12x^{2}-8-\left(x^{6}+8\right)$$

To find the opposite of x^{6}+8, find the opposite of each term.

$$x^{6}-6x^{4}+12x^{2}-8-x^{6}-8$$

Combine x^{6} and -x^{6} to get 0.

$$-6x^{4}+12x^{2}-8-8$$

Subtract 8 from -8 to get -16.

$$-6x^{4}+12x^{2}-16$$

Expand

$$-6x^{4}+12x^{2}-16$$

Solution Steps

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

$$\left(x^{2}\right)^{3}-6\left(x^{2}\right)^{2}+12x^{2}-8-\left(x^{6}+8\right)$$

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

$$x^{6}-6\left(x^{2}\right)^{2}+12x^{2}-8-\left(x^{6}+8\right)$$

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

$$x^{6}-6x^{4}+12x^{2}-8-\left(x^{6}+8\right)$$

To find the opposite of x^{6}+8, find the opposite of each term.

$$x^{6}-6x^{4}+12x^{2}-8-x^{6}-8$$

Combine x^{6} and -x^{6} to get 0.

$$-6x^{4}+12x^{2}-8-8$$

Subtract 8 from -8 to get -16.

$$-6x^{4}+12x^{2}-16$$