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

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Add -64 and 64 to get 0.

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

Expand

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

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Add -64 and 64 to get 0.

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