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

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

Evaluate

$$\frac{8\left(\sqrt{3}+2\right)}{3}\approx 9.952135487$$

Short Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Add 1 and 3 to get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3}{3}.

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

Since \frac{\sqrt{3}}{3} and \frac{3}{3} have the same denominator, add them by adding their numerators.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4+2\sqrt{3} times \frac{3^{2}}{3^{2}}.

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

Since \frac{\left(4+2\sqrt{3}\right)\times 3^{2}}{3^{2}} and \frac{\left(\sqrt{3}+3\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(4+2\sqrt{3}\right)\times 3^{2}+\left(\sqrt{3}+3\right)^{2}.

$$\frac{36+18\sqrt{3}+\left(\sqrt{3}\right)^{2}+6\sqrt{3}+9}{3^{2}}$$

Do the calculations in 36+18\sqrt{3}+\left(\sqrt{3}\right)^{2}+6\sqrt{3}+9.

$$\frac{48+24\sqrt{3}}{3^{2}}$$

Expand 3^{2}.

$$\frac{48+24\sqrt{3}}{9}$$

Expand

$$\frac{8 \sqrt{3} + 16}{3} = 9.95213548685034$$

Short Solution Steps

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

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

The square of \sqrt{3} is 3.

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

Add 1 and 3 to get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3}{3}.

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

Since \frac{\sqrt{3}}{3} and \frac{3}{3} have the same denominator, add them by adding their numerators.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4+2\sqrt{3} times \frac{3^{2}}{3^{2}}.

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

Since \frac{\left(4+2\sqrt{3}\right)\times 3^{2}}{3^{2}} and \frac{\left(\sqrt{3}+3\right)^{2}}{3^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(4+2\sqrt{3}\right)\times 3^{2}+\left(\sqrt{3}+3\right)^{2}.

$$\frac{36+18\sqrt{3}+\left(\sqrt{3}\right)^{2}+6\sqrt{3}+9}{3^{2}}$$

Do the calculations in 36+18\sqrt{3}+\left(\sqrt{3}\right)^{2}+6\sqrt{3}+9.

$$\frac{48+24\sqrt{3}}{3^{2}}$$

Expand 3^{2}.

$$\frac{48+24\sqrt{3}}{9}$$