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

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

Evaluate

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

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(\frac{1}{4}a^{2}+1\right)^{2}.

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

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

$$\frac{1}{16}a^{4}+\frac{1}{2}a^{2}+1-a^{2}$$

Combine \frac{1}{2}a^{2} and -a^{2} to get -\frac{1}{2}a^{2}.

$$\frac{1}{16}a^{4}-\frac{1}{2}a^{2}+1$$

Expand

$$\frac{a^{4}}{16}-\frac{a^{2}}{2}+1$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(\frac{1}{4}a^{2}+1\right)^{2}.

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

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

$$\frac{1}{16}a^{4}+\frac{1}{2}a^{2}+1-a^{2}$$

Combine \frac{1}{2}a^{2} and -a^{2} to get -\frac{1}{2}a^{2}.

$$\frac{1}{16}a^{4}-\frac{1}{2}a^{2}+1$$