Solve (3a^{2}-b^{2})^2-(a^2-3b^2)^2

$$( 3 a ^ { 2 } - b ^ { 2 } ) ^ { 2 } - ( a ^ { 2 } - 3 b ^ { 2 } ) ^ { 2 }$$

Evaluate

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

Solution Steps

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

$$9\left(a^{2}\right)^{2}-6a^{2}b^{2}+\left(b^{2}\right)^{2}-\left(a^{2}-3b^{2}\right)^{2}$$

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

$$9a^{4}-6a^{2}b^{2}+\left(b^{2}\right)^{2}-\left(a^{2}-3b^{2}\right)^{2}$$

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

$$9a^{4}-6a^{2}b^{2}+b^{4}-\left(a^{2}-3b^{2}\right)^{2}$$

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

$$9a^{4}-6a^{2}b^{2}+b^{4}-\left(\left(a^{2}\right)^{2}-6a^{2}b^{2}+9\left(b^{2}\right)^{2}\right)$$

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

$$9a^{4}-6a^{2}b^{2}+b^{4}-\left(a^{4}-6a^{2}b^{2}+9\left(b^{2}\right)^{2}\right)$$

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

$$9a^{4}-6a^{2}b^{2}+b^{4}-\left(a^{4}-6a^{2}b^{2}+9b^{4}\right)$$

To find the opposite of a^{4}-6a^{2}b^{2}+9b^{4}, find the opposite of each term.

$$9a^{4}-6a^{2}b^{2}+b^{4}-a^{4}+6a^{2}b^{2}-9b^{4}$$

Combine 9a^{4} and -a^{4} to get 8a^{4}.

$$8a^{4}-6a^{2}b^{2}+b^{4}+6a^{2}b^{2}-9b^{4}$$

Combine -6a^{2}b^{2} and 6a^{2}b^{2} to get 0.

$$8a^{4}+b^{4}-9b^{4}$$

Combine b^{4} and -9b^{4} to get -8b^{4}.

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

Expand

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

Solution Steps

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

$$9\left(a^{2}\right)^{2}-6a^{2}b^{2}+\left(b^{2}\right)^{2}-\left(a^{2}-3b^{2}\right)^{2}$$

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

$$9a^{4}-6a^{2}b^{2}+\left(b^{2}\right)^{2}-\left(a^{2}-3b^{2}\right)^{2}$$

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

$$9a^{4}-6a^{2}b^{2}+b^{4}-\left(a^{2}-3b^{2}\right)^{2}$$

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

$$9a^{4}-6a^{2}b^{2}+b^{4}-\left(\left(a^{2}\right)^{2}-6a^{2}b^{2}+9\left(b^{2}\right)^{2}\right)$$

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

$$9a^{4}-6a^{2}b^{2}+b^{4}-\left(a^{4}-6a^{2}b^{2}+9\left(b^{2}\right)^{2}\right)$$

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

$$9a^{4}-6a^{2}b^{2}+b^{4}-\left(a^{4}-6a^{2}b^{2}+9b^{4}\right)$$

To find the opposite of a^{4}-6a^{2}b^{2}+9b^{4}, find the opposite of each term.

$$9a^{4}-6a^{2}b^{2}+b^{4}-a^{4}+6a^{2}b^{2}-9b^{4}$$

Combine 9a^{4} and -a^{4} to get 8a^{4}.

$$8a^{4}-6a^{2}b^{2}+b^{4}+6a^{2}b^{2}-9b^{4}$$

Combine -6a^{2}b^{2} and 6a^{2}b^{2} to get 0.

$$8a^{4}+b^{4}-9b^{4}$$

Combine b^{4} and -9b^{4} to get -8b^{4}.

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