Solve 1+frac{(sqrt{6}+sqrt{2})^2}{4}+2frac{sqrt{6}+sqrt{2}}{2}*frac{sqrt{2}}{2}

$$1 + \frac { ( \sqrt { 6 } + \sqrt { 2 } ) ^ { 2 } } { 4 } + 2 \frac { \sqrt { 6 } + \sqrt { 2 } } { 2 } \cdot \frac { \sqrt { 2 } } { 2 }$$

Evaluate

$$2\left(\sqrt{3}+2\right)\approx 7.464101615$$

Short Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{6}+\sqrt{2}\right)^{2}.

$$1+\frac{\left(\sqrt{6}\right)^{2}+2\sqrt{6}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{4}+2\times \frac{\sqrt{6}+\sqrt{2}}{2}\times \frac{\sqrt{2}}{2}$$

The square of \sqrt{6} is 6.

$$1+\frac{6+2\sqrt{6}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{4}+2\times \frac{\sqrt{6}+\sqrt{2}}{2}\times \frac{\sqrt{2}}{2}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$1+\frac{6+2\sqrt{2}\sqrt{3}\sqrt{2}+\left(\sqrt{2}\right)^{2}}{4}+2\times \frac{\sqrt{6}+\sqrt{2}}{2}\times \frac{\sqrt{2}}{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$1+\frac{6+2\times 2\sqrt{3}+\left(\sqrt{2}\right)^{2}}{4}+2\times \frac{\sqrt{6}+\sqrt{2}}{2}\times \frac{\sqrt{2}}{2}$$

Multiply 2 and 2 to get 4.

$$1+\frac{6+4\sqrt{3}+\left(\sqrt{2}\right)^{2}}{4}+2\times \frac{\sqrt{6}+\sqrt{2}}{2}\times \frac{\sqrt{2}}{2}$$

The square of \sqrt{2} is 2.

$$1+\frac{6+4\sqrt{3}+2}{4}+2\times \frac{\sqrt{6}+\sqrt{2}}{2}\times \frac{\sqrt{2}}{2}$$

Add 6 and 2 to get 8.

$$1+\frac{8+4\sqrt{3}}{4}+2\times \frac{\sqrt{6}+\sqrt{2}}{2}\times \frac{\sqrt{2}}{2}$$

Divide each term of 8+4\sqrt{3} by 4 to get 2+\sqrt{3}.

$$1+2+\sqrt{3}+2\times \frac{\sqrt{6}+\sqrt{2}}{2}\times \frac{\sqrt{2}}{2}$$

Add 1 and 2 to get 3.

$$3+\sqrt{3}+2\times \frac{\sqrt{6}+\sqrt{2}}{2}\times \frac{\sqrt{2}}{2}$$

Cancel out 2 and 2.

$$3+\sqrt{3}+\left(\sqrt{6}+\sqrt{2}\right)\times \frac{\sqrt{2}}{2}$$

Express \left(\sqrt{6}+\sqrt{2}\right)\times \frac{\sqrt{2}}{2} as a single fraction.

$$3+\sqrt{3}+\frac{\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3+\sqrt{3} times \frac{2}{2}.

$$\frac{2\left(3+\sqrt{3}\right)}{2}+\frac{\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2}}{2}$$

Since \frac{2\left(3+\sqrt{3}\right)}{2} and \frac{\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2}}{2} have the same denominator, add them by adding their numerators.

$$\frac{2\left(3+\sqrt{3}\right)+\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2}}{2}$$

Do the multiplications in 2\left(3+\sqrt{3}\right)+\left(\sqrt{6}+\sqrt{2}\right)\sqrt{2}.

$$\frac{6+2\sqrt{3}+2\sqrt{3}+2}{2}$$

Do the calculations in 6+2\sqrt{3}+2\sqrt{3}+2.

$$\frac{8+4\sqrt{3}}{2}$$

Factor

$$2 {(\sqrt{3} + 2)} = 7.464101615$$