Solve {left(13/2+frac{sqrt{165}}{2}right)}^2

$${ \left( \frac{ 13 }{ 2 } + \frac{ \sqrt{ 165 } }{ 2 } \right) }^{ 2 }$$

Evaluate

$$\frac{13\sqrt{165}+167}{2}\approx 166.994011761$$

Solution Steps

Since \frac{13}{2} and \frac{\sqrt{165}}{2} have the same denominator, add them by adding their numerators.

$$\left(\frac{13+\sqrt{165}}{2}\right)^{2}$$

To raise \frac{13+\sqrt{165}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(13+\sqrt{165}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(13+\sqrt{165}\right)^{2}.

$$\frac{169+26\sqrt{165}+\left(\sqrt{165}\right)^{2}}{2^{2}}$$

The square of \sqrt{165} is 165.

$$\frac{169+26\sqrt{165}+165}{2^{2}}$$

Add 169 and 165 to get 334.

$$\frac{334+26\sqrt{165}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{334+26\sqrt{165}}{4}$$

Expand

$$\frac{13 \sqrt{165} + 167}{2} = 166.99401176132335$$

Solution Steps

Since \frac{13}{2} and \frac{\sqrt{165}}{2} have the same denominator, add them by adding their numerators.

$$\left(\frac{13+\sqrt{165}}{2}\right)^{2}$$

To raise \frac{13+\sqrt{165}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(13+\sqrt{165}\right)^{2}}{2^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(13+\sqrt{165}\right)^{2}.

$$\frac{169+26\sqrt{165}+\left(\sqrt{165}\right)^{2}}{2^{2}}$$

The square of \sqrt{165} is 165.

$$\frac{169+26\sqrt{165}+165}{2^{2}}$$

Add 169 and 165 to get 334.

$$\frac{334+26\sqrt{165}}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{334+26\sqrt{165}}{4}$$