Solve frac{2^2+(4sqrt{3}+1)^2-(-4)^2}{2*2*(5+1)}

$$\frac { 2 ^ { 2 } + ( 4 \sqrt { 3 } + 1 ) ^ { 2 } - ( - 4 ) ^ { 2 } } { 2 \times 2 \times ( 5 + 1 ) }$$

Evaluate

$$\frac{\sqrt{3}}{3}+\frac{37}{24}\approx 2.119016936$$

Solution Steps

Calculate 2 to the power of 2 and get 4.

$$\frac{4+\left(4\sqrt{3}+1\right)^{2}-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4\sqrt{3}+1\right)^{2}.

$$\frac{4+16\left(\sqrt{3}\right)^{2}+8\sqrt{3}+1-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

The square of \sqrt{3} is 3.

$$\frac{4+16\times 3+8\sqrt{3}+1-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Multiply 16 and 3 to get 48.

$$\frac{4+48+8\sqrt{3}+1-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Add 48 and 1 to get 49.

$$\frac{4+49+8\sqrt{3}-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Add 4 and 49 to get 53.

$$\frac{53+8\sqrt{3}-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{53+8\sqrt{3}-16}{2\times 2\left(5+1\right)}$$

Subtract 16 from 53 to get 37.

$$\frac{37+8\sqrt{3}}{2\times 2\left(5+1\right)}$$

Multiply 2 and 2 to get 4.

$$\frac{37+8\sqrt{3}}{4\left(5+1\right)}$$

Add 5 and 1 to get 6.

$$\frac{37+8\sqrt{3}}{4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{37+8\sqrt{3}}{24}$$

Expand

$$\frac{\sqrt{3}}{3} + \frac{37}{24} = 2.119016936$$

Solution Steps

Calculate 2 to the power of 2 and get 4.

$$\frac{4+\left(4\sqrt{3}+1\right)^{2}-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4\sqrt{3}+1\right)^{2}.

$$\frac{4+16\left(\sqrt{3}\right)^{2}+8\sqrt{3}+1-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

The square of \sqrt{3} is 3.

$$\frac{4+16\times 3+8\sqrt{3}+1-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Multiply 16 and 3 to get 48.

$$\frac{4+48+8\sqrt{3}+1-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Add 48 and 1 to get 49.

$$\frac{4+49+8\sqrt{3}-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Add 4 and 49 to get 53.

$$\frac{53+8\sqrt{3}-\left(-4\right)^{2}}{2\times 2\left(5+1\right)}$$

Calculate -4 to the power of 2 and get 16.

$$\frac{53+8\sqrt{3}-16}{2\times 2\left(5+1\right)}$$

Subtract 16 from 53 to get 37.

$$\frac{37+8\sqrt{3}}{2\times 2\left(5+1\right)}$$

Multiply 2 and 2 to get 4.

$$\frac{37+8\sqrt{3}}{4\left(5+1\right)}$$

Add 5 and 1 to get 6.

$$\frac{37+8\sqrt{3}}{4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{37+8\sqrt{3}}{24}$$