Solve [(a^{8}-a^{12}-a^{16}):(-1/2a^{8})+2]^{2}:[-2(-a)^{4}]^2-(-a^2)^2(a^4+2)

$$[ ( a ^ { 8 } - a ^ { 12 } - a ^ { 16 } ) : ( - \frac { 1 } { 2 } a ^ { 8 } ) + 2 ] ^ { 2 } : [ - 2 ( - a ) ^ { 4 } ] ^ { 2 } - ( - a ^ { 2 } ) ^ { 2 } ( a ^ { 4 } + 2 )$$

Evaluate

$$1$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{8}-a^{12}-a^{16}}{-\frac{1}{2}a^{8}}.

$$\frac{\left(\frac{-\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)a^{8}}{-\frac{1}{2}a^{8}}+2\right)^{2}}{\left(-2\left(-a\right)^{4}\right)^{2}}-\left(-a^{2}\right)^{2}\left(a^{4}+2\right)$$

Cancel out a^{8} in both numerator and denominator.

$$\frac{\left(\frac{-\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)}{-\frac{1}{2}}+2\right)^{2}}{\left(-2\left(-a\right)^{4}\right)^{2}}-\left(-a^{2}\right)^{2}\left(a^{4}+2\right)$$

Divide -\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right) by -\frac{1}{2} to get 2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right).

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{\left(-2\left(-a\right)^{4}\right)^{2}}-\left(-a^{2}\right)^{2}\left(a^{4}+2\right)$$

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

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{\left(-2\right)^{2}\left(\left(-a\right)^{4}\right)^{2}}-\left(-a^{2}\right)^{2}\left(a^{4}+2\right)$$

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

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{\left(-2\right)^{2}\left(-a\right)^{8}}-\left(-a^{2}\right)^{2}\left(a^{4}+2\right)$$

Calculate -2 to the power of 2 and get 4.

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{4\left(-a\right)^{8}}-\left(-a^{2}\right)^{2}\left(a^{4}+2\right)$$

Calculate -a^{2} to the power of 2 and get \left(a^{2}\right)^{2}.

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{4\left(-a\right)^{8}}-\left(a^{2}\right)^{2}\left(a^{4}+2\right)$$

Use the distributive property to multiply \left(a^{2}\right)^{2} by a^{4}+2.

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{4\left(-a\right)^{8}}-\left(\left(a^{2}\right)^{2}a^{4}+2\left(a^{2}\right)^{2}\right)$$

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

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{4\left(-a\right)^{8}}-\left(a^{4}a^{4}+2\left(a^{2}\right)^{2}\right)$$

To multiply powers of the same base, add their exponents. Add 4 and 4 to get 8.

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{4\left(-a\right)^{8}}-\left(a^{8}+2\left(a^{2}\right)^{2}\right)$$

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

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{4\left(-a\right)^{8}}-\left(a^{8}+2a^{4}\right)$$

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

$$\frac{\left(2\left(a^{4}-\left(-\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{4\left(-a\right)^{8}}-a^{8}-2a^{4}$$

To find the opposite of -\frac{1}{2}\sqrt{5}-\frac{1}{2}, find the opposite of each term.

$$\frac{\left(2\left(a^{4}+\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\left(a^{4}-\left(\frac{1}{2}\sqrt{5}-\frac{1}{2}\right)\right)+2\right)^{2}}{4\left(-a\right)^{8}}-a^{8}-2a^{4}$$

To find the opposite of \frac{1}{2}\sqrt{5}-\frac{1}{2}, find the opposite of each term.

$$\frac{\left(2\left(a^{4}+\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)\left(a^{4}-\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)+2\right)^{2}}{4\left(-a\right)^{8}}-a^{8}-2a^{4}$$

Use the distributive property to multiply 2 by a^{4}+\frac{1}{2}\sqrt{5}+\frac{1}{2}.

$$\frac{\left(\left(2a^{4}+\sqrt{5}+1\right)\left(a^{4}-\frac{1}{2}\sqrt{5}+\frac{1}{2}\right)+2\right)^{2}}{4\left(-a\right)^{8}}-a^{8}-2a^{4}$$

Use the distributive property to multiply 2a^{4}+\sqrt{5}+1 by a^{4}-\frac{1}{2}\sqrt{5}+\frac{1}{2} and combine like terms.

$$\frac{\left(2a^{8}+2a^{4}-\frac{1}{2}\left(\sqrt{5}\right)^{2}+\frac{1}{2}+2\right)^{2}}{4\left(-a\right)^{8}}-a^{8}-2a^{4}$$

The square of \sqrt{5} is 5.

$$\frac{\left(2a^{8}+2a^{4}-\frac{1}{2}\times 5+\frac{1}{2}+2\right)^{2}}{4\left(-a\right)^{8}}-a^{8}-2a^{4}$$

Multiply -\frac{1}{2} and 5 to get -\frac{5}{2}.

$$\frac{\left(2a^{8}+2a^{4}-\frac{5}{2}+\frac{1}{2}+2\right)^{2}}{4\left(-a\right)^{8}}-a^{8}-2a^{4}$$

Add -\frac{5}{2} and \frac{1}{2} to get -2.

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

Add -2 and 2 to get 0.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -a^{8}-2a^{4} times \frac{4\left(-a\right)^{8}}{4\left(-a\right)^{8}}.

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

Since \frac{\left(2a^{8}+2a^{4}\right)^{2}}{4\left(-a\right)^{8}} and \frac{\left(-a^{8}-2a^{4}\right)\times 4\left(-a\right)^{8}}{4\left(-a\right)^{8}} have the same denominator, add them by adding their numerators.

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

Expand \left(-a\right)^{8}.

$$\frac{\left(2a^{8}+2a^{4}\right)^{2}}{4\left(-1\right)^{8}a^{8}}-a^{8}-2a^{4}$$

Calculate -1 to the power of 8 and get 1.

$$\frac{\left(2a^{8}+2a^{4}\right)^{2}}{4\times 1a^{8}}-a^{8}-2a^{4}$$

Multiply 4 and 1 to get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -a^{8}-2a^{4} times \frac{4a^{8}}{4a^{8}}.

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

Since \frac{\left(2a^{8}+2a^{4}\right)^{2}}{4a^{8}} and \frac{\left(-a^{8}-2a^{4}\right)\times 4a^{8}}{4a^{8}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(2a^{8}+2a^{4}\right)^{2}+\left(-a^{8}-2a^{4}\right)\times 4a^{8}.

$$\frac{4a^{16}+8a^{12}+4a^{8}-4a^{16}-8a^{12}}{4a^{8}}$$

Combine like terms in 4a^{16}+8a^{12}+4a^{8}-4a^{16}-8a^{12}.

$$\frac{4a^{8}}{4a^{8}}$$

Cancel out 4a^{8} in both numerator and denominator.

$$1$$

Factor

$$1$$