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

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

Evaluate

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

Solution Steps

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

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

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

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

The square of \sqrt{5} is 5.

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

Add 5 and 9 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{\sqrt{5}+3}{2} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

The square of \sqrt{5} is 5.

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

Add 5 and 9 to get 14.

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

Calculate 2 to the power of 2 and get 4.

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