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

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

Evaluate

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

Solution Steps

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

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

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

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

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

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

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}.

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

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

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

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

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

To find the opposite of 64-48y^{2}+12y^{4}-y^{6}, find the opposite of each term.

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

Subtract 64 from 64 to get 0.

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

Expand

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

Solution Steps

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

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

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

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

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

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

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}.

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

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

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

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

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

To find the opposite of 64-48y^{2}+12y^{4}-y^{6}, find the opposite of each term.

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

Subtract 64 from 64 to get 0.

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