Solve {left(frac{3+sqrt{5}}{2}right)}^2

$${ \left( \frac{ 3+ \sqrt{ 5 } }{ 2 } \right) }^{ 2 }$$

Evaluate

$$\frac{3\sqrt{5}+7}{2}\approx 6.854101966$$

Solution Steps

To raise \frac{3+\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(3+\sqrt{5}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{5}\right)^{2}.

$$\frac{9+6\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{9+6\sqrt{5}+5}{2^{2}}$$

Add 9 and 5 to get 14.

$$\frac{14+6\sqrt{5}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{14+6\sqrt{5}}{4}$$

Expand

$$\frac{3 \sqrt{5} + 7}{2} = 6.854101966249685$$

Solution Steps

To raise \frac{3+\sqrt{5}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(3+\sqrt{5}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{5}\right)^{2}.

$$\frac{9+6\sqrt{5}+\left(\sqrt{5}\right)^{2}}{2^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{9+6\sqrt{5}+5}{2^{2}}$$

Add 9 and 5 to get 14.

$$\frac{14+6\sqrt{5}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{14+6\sqrt{5}}{4}$$