Solve (frac{-1+sqrt{145}}{6})^2+(frac{-1-sqrt{145}}{6})^2

$$( \frac { - 1 + \sqrt { 145 } } { 6 } ) ^ { 2 } + ( \frac { - 1 - \sqrt { 145 } } { 6 } ) ^ { 2 }$$

Evaluate

$$\frac{73}{9}\approx 8.111111111$$

Short Solution Steps

To raise \frac{-1+\sqrt{145}}{6} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-1+\sqrt{145}\right)^{2}}{6^{2}}+\left(\frac{-1-\sqrt{145}}{6}\right)^{2}$$

To raise \frac{-1-\sqrt{145}}{6} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(-1+\sqrt{145}\right)^{2}}{6^{2}}+\frac{\left(-1-\sqrt{145}\right)^{2}}{6^{2}}$$

Since \frac{\left(-1+\sqrt{145}\right)^{2}}{6^{2}} and \frac{\left(-1-\sqrt{145}\right)^{2}}{6^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(-1+\sqrt{145}\right)^{2}+\left(-1-\sqrt{145}\right)^{2}}{6^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-1+\sqrt{145}\right)^{2}.

$$\frac{1-2\sqrt{145}+\left(\sqrt{145}\right)^{2}}{6^{2}}+\frac{\left(-1-\sqrt{145}\right)^{2}}{6^{2}}$$

The square of \sqrt{145} is 145.

$$\frac{1-2\sqrt{145}+145}{6^{2}}+\frac{\left(-1-\sqrt{145}\right)^{2}}{6^{2}}$$

Add 1 and 145 to get 146.

$$\frac{146-2\sqrt{145}}{6^{2}}+\frac{\left(-1-\sqrt{145}\right)^{2}}{6^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{146-2\sqrt{145}}{36}+\frac{\left(-1-\sqrt{145}\right)^{2}}{6^{2}}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-1-\sqrt{145}\right)^{2}.

$$\frac{146-2\sqrt{145}}{36}+\frac{1+2\sqrt{145}+\left(\sqrt{145}\right)^{2}}{6^{2}}$$

The square of \sqrt{145} is 145.

$$\frac{146-2\sqrt{145}}{36}+\frac{1+2\sqrt{145}+145}{6^{2}}$$

Add 1 and 145 to get 146.

$$\frac{146-2\sqrt{145}}{36}+\frac{146+2\sqrt{145}}{6^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{146-2\sqrt{145}}{36}+\frac{146+2\sqrt{145}}{36}$$

Since \frac{146-2\sqrt{145}}{36} and \frac{146+2\sqrt{145}}{36} have the same denominator, add them by adding their numerators.

$$\frac{146-2\sqrt{145}+146+2\sqrt{145}}{36}$$

Do the calculations in 146-2\sqrt{145}+146+2\sqrt{145}.

$$\frac{292}{36}$$

Factor

$$\frac{73}{3 ^ {2}} = 8\frac{1}{9} = 8.11111111111111$$