Solve {left(frac{sqrt{3}+1}{2}right)}^2+{left(frac{sqrt{3}+3}{2}right)}^2

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

Evaluate

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

Short Solution Steps

To raise \frac{\sqrt{3}+1}{2} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{\sqrt{3}+3}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Since \frac{\left(\sqrt{3}+1\right)^{2}}{2^{2}} and \frac{\left(\sqrt{3}+3\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(\sqrt{3}+1\right)^{2}+\left(\sqrt{3}+3\right)^{2}.

$$\frac{\left(\sqrt{3}\right)^{2}+2\sqrt{3}+1+\left(\sqrt{3}\right)^{2}+6\sqrt{3}+9}{2^{2}}$$

Do the calculations in \left(\sqrt{3}\right)^{2}+2\sqrt{3}+1+\left(\sqrt{3}\right)^{2}+6\sqrt{3}+9.

$$\frac{16+8\sqrt{3}}{2^{2}}$$

Expand 2^{2}.

$$\frac{16+8\sqrt{3}}{4}$$

Expand

$$2 \sqrt{3} + 4 = 7.464101615$$

Short Solution Steps

To raise \frac{\sqrt{3}+1}{2} to a power, raise both numerator and denominator to the power and then divide.

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

To raise \frac{\sqrt{3}+3}{2} to a power, raise both numerator and denominator to the power and then divide.

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

Since \frac{\left(\sqrt{3}+1\right)^{2}}{2^{2}} and \frac{\left(\sqrt{3}+3\right)^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(\sqrt{3}+1\right)^{2}+\left(\sqrt{3}+3\right)^{2}.

$$\frac{\left(\sqrt{3}\right)^{2}+2\sqrt{3}+1+\left(\sqrt{3}\right)^{2}+6\sqrt{3}+9}{2^{2}}$$

Do the calculations in \left(\sqrt{3}\right)^{2}+2\sqrt{3}+1+\left(\sqrt{3}\right)^{2}+6\sqrt{3}+9.

$$\frac{16+8\sqrt{3}}{2^{2}}$$

Expand 2^{2}.

$$\frac{16+8\sqrt{3}}{4}$$