Solve (frac{1+sqrt{5}}{4})^2

$$( \frac { 1 + \sqrt { 5 } } { 4 } ) ^ { 2 }$$

Evaluate

$$\frac{\sqrt{5}+3}{8}\approx 0.654508497$$

Solution Steps

To raise \frac{1+\sqrt{5}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(1+\sqrt{5}\right)^{2}}{4^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{5}\right)^{2}.

$$\frac{1+2\sqrt{5}+\left(\sqrt{5}\right)^{2}}{4^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{1+2\sqrt{5}+5}{4^{2}}$$

Add 1 and 5 to get 6.

$$\frac{6+2\sqrt{5}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{6+2\sqrt{5}}{16}$$

Expand

$$\frac{\sqrt{5} + 3}{8} = 0.6545084971874737$$

Solution Steps

To raise \frac{1+\sqrt{5}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(1+\sqrt{5}\right)^{2}}{4^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{5}\right)^{2}.

$$\frac{1+2\sqrt{5}+\left(\sqrt{5}\right)^{2}}{4^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{1+2\sqrt{5}+5}{4^{2}}$$

Add 1 and 5 to get 6.

$$\frac{6+2\sqrt{5}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{6+2\sqrt{5}}{16}$$