Solve (3a-2b)(3a+2b)(9a^{2}+4b^2)-(-3a)^4+(-3b^2)^2

$$( 3 a - 2 b ) ( 3 a + 2 b ) ( 9 a ^ { 2 } + 4 b ^ { 2 } ) - ( - 3 a ) ^ { 4 } + ( - 3 b ^ { 2 } ) ^ { 2 }$$

Evaluate

$$-7b^{4}$$

Solution Steps

Use the distributive property to multiply 3a-2b by 3a+2b and combine like terms.

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

Consider \left(9a^{2}-4b^{2}\right)\left(9a^{2}+4b^{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(9a^{2}\right)^{2}.

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

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

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

Calculate 9 to the power of 2 and get 81.

$$81a^{4}-\left(4b^{2}\right)^{2}-\left(-3a\right)^{4}+\left(-3b^{2}\right)^{2}$$

Expand \left(4b^{2}\right)^{2}.

$$81a^{4}-4^{2}\left(b^{2}\right)^{2}-\left(-3a\right)^{4}+\left(-3b^{2}\right)^{2}$$

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

$$81a^{4}-4^{2}b^{4}-\left(-3a\right)^{4}+\left(-3b^{2}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$81a^{4}-16b^{4}-\left(-3a\right)^{4}+\left(-3b^{2}\right)^{2}$$

Expand \left(-3a\right)^{4}.

$$81a^{4}-16b^{4}-\left(-3\right)^{4}a^{4}+\left(-3b^{2}\right)^{2}$$

Calculate -3 to the power of 4 and get 81.

$$81a^{4}-16b^{4}-81a^{4}+\left(-3b^{2}\right)^{2}$$

Combine 81a^{4} and -81a^{4} to get 0.

$$-16b^{4}+\left(-3b^{2}\right)^{2}$$

Expand \left(-3b^{2}\right)^{2}.

$$-16b^{4}+\left(-3\right)^{2}\left(b^{2}\right)^{2}$$

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

$$-16b^{4}+\left(-3\right)^{2}b^{4}$$

Calculate -3 to the power of 2 and get 9.

$$-16b^{4}+9b^{4}$$

Combine -16b^{4} and 9b^{4} to get -7b^{4}.

$$-7b^{4}$$

Expand

$$-7b^{4}$$

Solution Steps

Use the distributive property to multiply 3a-2b by 3a+2b and combine like terms.

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

Consider \left(9a^{2}-4b^{2}\right)\left(9a^{2}+4b^{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(9a^{2}\right)^{2}.

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

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

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

Calculate 9 to the power of 2 and get 81.

$$81a^{4}-\left(4b^{2}\right)^{2}-\left(-3a\right)^{4}+\left(-3b^{2}\right)^{2}$$

Expand \left(4b^{2}\right)^{2}.

$$81a^{4}-4^{2}\left(b^{2}\right)^{2}-\left(-3a\right)^{4}+\left(-3b^{2}\right)^{2}$$

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

$$81a^{4}-4^{2}b^{4}-\left(-3a\right)^{4}+\left(-3b^{2}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$81a^{4}-16b^{4}-\left(-3a\right)^{4}+\left(-3b^{2}\right)^{2}$$

Expand \left(-3a\right)^{4}.

$$81a^{4}-16b^{4}-\left(-3\right)^{4}a^{4}+\left(-3b^{2}\right)^{2}$$

Calculate -3 to the power of 4 and get 81.

$$81a^{4}-16b^{4}-81a^{4}+\left(-3b^{2}\right)^{2}$$

Combine 81a^{4} and -81a^{4} to get 0.

$$-16b^{4}+\left(-3b^{2}\right)^{2}$$

Expand \left(-3b^{2}\right)^{2}.

$$-16b^{4}+\left(-3\right)^{2}\left(b^{2}\right)^{2}$$

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

$$-16b^{4}+\left(-3\right)^{2}b^{4}$$

Calculate -3 to the power of 2 and get 9.

$$-16b^{4}+9b^{4}$$

Combine -16b^{4} and 9b^{4} to get -7b^{4}.

$$-7b^{4}$$