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

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

Evaluate

$$\frac{4\sqrt{5}}{9}+1\approx 1.99380799$$

Solution Steps

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

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

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

$$\frac{\left(\sqrt{5}\right)^{2}+4\sqrt{5}+4}{3^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{5+4\sqrt{5}+4}{3^{2}}$$

Add 5 and 4 to get 9.

$$\frac{9+4\sqrt{5}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9+4\sqrt{5}}{9}$$

Expand

$$\frac{4 \sqrt{5}}{9} + 1 = 1.99380799$$

Solution Steps

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

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

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

$$\frac{\left(\sqrt{5}\right)^{2}+4\sqrt{5}+4}{3^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{5+4\sqrt{5}+4}{3^{2}}$$

Add 5 and 4 to get 9.

$$\frac{9+4\sqrt{5}}{3^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9+4\sqrt{5}}{9}$$