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

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

Evaluate

$$\frac{81\left(\sqrt{5}+3\right)}{8}\approx 53.015188272$$

Solution Steps

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

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

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

$$\frac{81+162\sqrt{5}+81\left(\sqrt{5}\right)^{2}}{4^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{81+162\sqrt{5}+81\times 5}{4^{2}}$$

Multiply 81 and 5 to get 405.

$$\frac{81+162\sqrt{5}+405}{4^{2}}$$

Add 81 and 405 to get 486.

$$\frac{486+162\sqrt{5}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{486+162\sqrt{5}}{16}$$

Expand

$$\frac{81 \sqrt{5} + 243}{8} = 53.01518827218537$$

Solution Steps

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

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

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

$$\frac{81+162\sqrt{5}+81\left(\sqrt{5}\right)^{2}}{4^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{81+162\sqrt{5}+81\times 5}{4^{2}}$$

Multiply 81 and 5 to get 405.

$$\frac{81+162\sqrt{5}+405}{4^{2}}$$

Add 81 and 405 to get 486.

$$\frac{486+162\sqrt{5}}{4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{486+162\sqrt{5}}{16}$$