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

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

Evaluate

$$\frac{\sqrt{3}}{2}+1\approx 1.866025404$$

Solution Steps

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

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

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

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

The square of \sqrt{3} is 3.

$$\frac{3+2\sqrt{3}+1}{2^{2}}$$

Add 3 and 1 to get 4.

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

Calculate 2 to the power of 2 and get 4.

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

Expand

$$\frac{\sqrt{3}}{2} + 1 = 1.866025404$$

Solution Steps

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

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

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

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

The square of \sqrt{3} is 3.

$$\frac{3+2\sqrt{3}+1}{2^{2}}$$

Add 3 and 1 to get 4.

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

Calculate 2 to the power of 2 and get 4.

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