Solve (a+b)*(a-b)*(a^{2}+b^{2})*(a^{4}+b^{4})-(a^{4}+b^{4})^2+2b^4*(b^4-a^4)

$$( a + b ) \cdot ( a - b ) \cdot ( a ^ { 2 } + b ^ { 2 } ) \cdot ( a ^ { 4 } + b ^ { 4 } ) - ( a ^ { 4 } + b ^ { 4 } ) ^ { 2 } + 2 b ^ { 4 } \cdot ( b ^ { 4 } - a ^ { 4 } )$$

Evaluate

$$-4\left(ab\right)^{4}$$

Solution Steps

Use the distributive property to multiply a+b by a-b and combine like terms.

$$\left(a^{2}-b^{2}\right)\left(a^{2}+b^{2}\right)\left(a^{4}+b^{4}\right)-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

Use the distributive property to multiply a^{2}-b^{2} by a^{2}+b^{2} and combine like terms.

$$\left(a^{4}-b^{4}\right)\left(a^{4}+b^{4}\right)-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

Consider \left(a^{4}-b^{4}\right)\left(a^{4}+b^{4}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(a^{4}\right)^{2}-\left(b^{4}\right)^{2}-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-\left(b^{4}\right)^{2}-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-b^{8}-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-b^{8}-\left(\left(a^{4}\right)^{2}+2a^{4}b^{4}+\left(b^{4}\right)^{2}\right)+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-b^{8}-\left(a^{8}+2a^{4}b^{4}+\left(b^{4}\right)^{2}\right)+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-b^{8}-\left(a^{8}+2a^{4}b^{4}+b^{8}\right)+2b^{4}\left(b^{4}-a^{4}\right)$$

To find the opposite of a^{8}+2a^{4}b^{4}+b^{8}, find the opposite of each term.

$$a^{8}-b^{8}-a^{8}-2a^{4}b^{4}-b^{8}+2b^{4}\left(b^{4}-a^{4}\right)$$

Combine a^{8} and -a^{8} to get 0.

$$-b^{8}-2a^{4}b^{4}-b^{8}+2b^{4}\left(b^{4}-a^{4}\right)$$

Combine -b^{8} and -b^{8} to get -2b^{8}.

$$-2b^{8}-2a^{4}b^{4}+2b^{4}\left(b^{4}-a^{4}\right)$$

Use the distributive property to multiply 2b^{4} by b^{4}-a^{4}.

$$-2b^{8}-2a^{4}b^{4}+2b^{8}-2b^{4}a^{4}$$

Combine -2b^{8} and 2b^{8} to get 0.

$$-2a^{4}b^{4}-2b^{4}a^{4}$$

Combine -2a^{4}b^{4} and -2b^{4}a^{4} to get -4a^{4}b^{4}.

$$-4a^{4}b^{4}$$

Expand

$$-4\left(ab\right)^{4}$$

Solution Steps

Use the distributive property to multiply a+b by a-b and combine like terms.

$$\left(a^{2}-b^{2}\right)\left(a^{2}+b^{2}\right)\left(a^{4}+b^{4}\right)-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

Use the distributive property to multiply a^{2}-b^{2} by a^{2}+b^{2} and combine like terms.

$$\left(a^{4}-b^{4}\right)\left(a^{4}+b^{4}\right)-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

Consider \left(a^{4}-b^{4}\right)\left(a^{4}+b^{4}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(a^{4}\right)^{2}-\left(b^{4}\right)^{2}-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-\left(b^{4}\right)^{2}-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-b^{8}-\left(a^{4}+b^{4}\right)^{2}+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-b^{8}-\left(\left(a^{4}\right)^{2}+2a^{4}b^{4}+\left(b^{4}\right)^{2}\right)+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-b^{8}-\left(a^{8}+2a^{4}b^{4}+\left(b^{4}\right)^{2}\right)+2b^{4}\left(b^{4}-a^{4}\right)$$

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

$$a^{8}-b^{8}-\left(a^{8}+2a^{4}b^{4}+b^{8}\right)+2b^{4}\left(b^{4}-a^{4}\right)$$

To find the opposite of a^{8}+2a^{4}b^{4}+b^{8}, find the opposite of each term.

$$a^{8}-b^{8}-a^{8}-2a^{4}b^{4}-b^{8}+2b^{4}\left(b^{4}-a^{4}\right)$$

Combine a^{8} and -a^{8} to get 0.

$$-b^{8}-2a^{4}b^{4}-b^{8}+2b^{4}\left(b^{4}-a^{4}\right)$$

Combine -b^{8} and -b^{8} to get -2b^{8}.

$$-2b^{8}-2a^{4}b^{4}+2b^{4}\left(b^{4}-a^{4}\right)$$

Use the distributive property to multiply 2b^{4} by b^{4}-a^{4}.

$$-2b^{8}-2a^{4}b^{4}+2b^{8}-2b^{4}a^{4}$$

Combine -2b^{8} and 2b^{8} to get 0.

$$-2a^{4}b^{4}-2b^{4}a^{4}$$

Combine -2a^{4}b^{4} and -2b^{4}a^{4} to get -4a^{4}b^{4}.

$$-4a^{4}b^{4}$$