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

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

Evaluate

$$\frac{15}{16}=0.9375$$

Solution Steps

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

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

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

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

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

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

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

$$-\frac{1}{16}+a^{4}+1-a^{4}$$

Add -\frac{1}{16} and 1 to get \frac{15}{16}.

$$\frac{15}{16}+a^{4}-a^{4}$$

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

$$\frac{15}{16}$$

Factor

$$\frac{3 \cdot 5}{2 ^ {4}} = 0.9375$$

Solution Steps

Factor out \frac{1}{16}.

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

Simplify.

$$\frac{15}{16}$$