Solve (1+-4a^2)(1+4a^2)-(a^2-4)(a^2+4)+15

$$( 1 + - 4 a ^ { 2 } ) ( 1 + 4 a ^ { 2 } ) - ( a ^ { 2 } - 4 ) ( a ^ { 2 } + 4 ) + 15$$

Evaluate

$$32-17a^{4}$$

Solution Steps

Consider \left(1-4a^{2}\right)\left(1+4a^{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}. Square 1.

$$1-\left(4a^{2}\right)^{2}-\left(a^{2}-4\right)\left(a^{2}+4\right)+15$$

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

$$1-4^{2}\left(a^{2}\right)^{2}-\left(a^{2}-4\right)\left(a^{2}+4\right)+15$$

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

$$1-4^{2}a^{4}-\left(a^{2}-4\right)\left(a^{2}+4\right)+15$$

Calculate 4 to the power of 2 and get 16.

$$1-16a^{4}-\left(a^{2}-4\right)\left(a^{2}+4\right)+15$$

Consider \left(a^{2}-4\right)\left(a^{2}+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}. Square 4.

$$1-16a^{4}-\left(\left(a^{2}\right)^{2}-16\right)+15$$

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

$$1-16a^{4}-\left(a^{4}-16\right)+15$$

To find the opposite of a^{4}-16, find the opposite of each term.

$$1-16a^{4}-a^{4}+16+15$$

Combine -16a^{4} and -a^{4} to get -17a^{4}.

$$1-17a^{4}+16+15$$

Add 1 and 16 to get 17.

$$17-17a^{4}+15$$

Add 17 and 15 to get 32.

$$32-17a^{4}$$

Expand

$$32-17a^{4}$$

Solution Steps

Consider \left(1-4a^{2}\right)\left(1+4a^{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}. Square 1.

$$1-\left(4a^{2}\right)^{2}-\left(a^{2}-4\right)\left(a^{2}+4\right)+15$$

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

$$1-4^{2}\left(a^{2}\right)^{2}-\left(a^{2}-4\right)\left(a^{2}+4\right)+15$$

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

$$1-4^{2}a^{4}-\left(a^{2}-4\right)\left(a^{2}+4\right)+15$$

Calculate 4 to the power of 2 and get 16.

$$1-16a^{4}-\left(a^{2}-4\right)\left(a^{2}+4\right)+15$$

Consider \left(a^{2}-4\right)\left(a^{2}+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}. Square 4.

$$1-16a^{4}-\left(\left(a^{2}\right)^{2}-16\right)+15$$

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

$$1-16a^{4}-\left(a^{4}-16\right)+15$$

To find the opposite of a^{4}-16, find the opposite of each term.

$$1-16a^{4}-a^{4}+16+15$$

Combine -16a^{4} and -a^{4} to get -17a^{4}.

$$1-17a^{4}+16+15$$

Add 1 and 16 to get 17.

$$17-17a^{4}+15$$

Add 17 and 15 to get 32.

$$32-17a^{4}$$