Solve (frac{9+2sqrt{5}}{4})^2

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

Evaluate

$$\frac{9\sqrt{5}}{4}+\frac{101}{16}\approx 11.343652949$$

Solution Steps

To raise \frac{9+2\sqrt{5}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(9+2\sqrt{5}\right)^{2}}{4^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9+2\sqrt{5}\right)^{2}.

$$\frac{81+36\sqrt{5}+4\left(\sqrt{5}\right)^{2}}{4^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{81+36\sqrt{5}+4\times 5}{4^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{81+36\sqrt{5}+20}{4^{2}}$$

Add 81 and 20 to get 101.

$$\frac{101+36\sqrt{5}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{101+36\sqrt{5}}{16}$$

Expand

$$\frac{9 \sqrt{5}}{4} + \frac{101}{16} = 11.343652949$$

Solution Steps

To raise \frac{9+2\sqrt{5}}{4} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(9+2\sqrt{5}\right)^{2}}{4^{2}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9+2\sqrt{5}\right)^{2}.

$$\frac{81+36\sqrt{5}+4\left(\sqrt{5}\right)^{2}}{4^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{81+36\sqrt{5}+4\times 5}{4^{2}}$$

Multiply 4 and 5 to get 20.

$$\frac{81+36\sqrt{5}+20}{4^{2}}$$

Add 81 and 20 to get 101.

$$\frac{101+36\sqrt{5}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{101+36\sqrt{5}}{16}$$